using symmetry to solve differential equationsmartinj/research/lewisclark_march2012.pdfusing...
TRANSCRIPT
![Page 1: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/1.jpg)
Using symmetry to solve differential equations
Martin Jackson
Mathematics and Computer Science, University of Puget Sound
March 6, 2012
![Page 2: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/2.jpg)
Outline
1 “Magic” coordinates
2 Symmetries of a differential equation
3 Using a symmetry to find “magic” coordinates
4 Finding symmetries of a differential equation
5 Topics for another time
![Page 3: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/3.jpg)
Outline
1 “Magic” coordinates
2 Symmetries of a differential equation
3 Using a symmetry to find “magic” coordinates
4 Finding symmetries of a differential equation
5 Topics for another time
![Page 4: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/4.jpg)
Outline
1 “Magic” coordinates
2 Symmetries of a differential equation
3 Using a symmetry to find “magic” coordinates
4 Finding symmetries of a differential equation
5 Topics for another time
![Page 5: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/5.jpg)
Outline
1 “Magic” coordinates
2 Symmetries of a differential equation
3 Using a symmetry to find “magic” coordinates
4 Finding symmetries of a differential equation
5 Topics for another time
![Page 6: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/6.jpg)
Outline
1 “Magic” coordinates
2 Symmetries of a differential equation
3 Using a symmetry to find “magic” coordinates
4 Finding symmetries of a differential equation
5 Topics for another time
![Page 7: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/7.jpg)
“Magic” coordinates
• start with a differential equationdy
dx= f (x , y)
• goal: find new variables to getds
dr= g(r)
dy
dx= f (x , y)
0 1 2 3 4
-2
-1
0
1
2
x
y
ds
dr= g(r)
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 8: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/8.jpg)
“Magic” coordinates
• start with a differential equationdy
dx= f (x , y)
• goal: find new variables to getds
dr= g(r)
dy
dx= f (x , y)
0 1 2 3 4
-2
-1
0
1
2
x
y
ds
dr= g(r)
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 9: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/9.jpg)
“Magic” coordinates
• start with a differential equationdy
dx= f (x , y)
• goal: find new variables to getds
dr= g(r)
dy
dx= f (x , y)
0 1 2 3 4
-2
-1
0
1
2
x
y
ds
dr= g(r)
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 10: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/10.jpg)
“Magic” coordinates
• start with a differential equationdy
dx= f (x , y)
• goal: find new variables to getds
dr= g(r)
dy
dx= f (x , y)
0 1 2 3 4
-2
-1
0
1
2
x
y
ds
dr= g(r)
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 11: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/11.jpg)
“Magic” coordinates
• start with a differential equationdy
dx= f (x , y)
• goal: find new variables to getds
dr= g(r)
dy
dx= f (x , y)
0 1 2 3 4
-2
-1
0
1
2
x
y
ds
dr= g(r)
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 12: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/12.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y=
1
r
![Page 13: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/13.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y=
1
r
![Page 14: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/14.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y=
1
r
![Page 15: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/15.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y=
1
r
![Page 16: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/16.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y=
1
r
![Page 17: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/17.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y=
1
r
![Page 18: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/18.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr
=dx
dy=
1
dy/dx=
1
y=
1
r
![Page 19: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/19.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy
=1
dy/dx=
1
y=
1
r
![Page 20: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/20.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx
=1
y=
1
r
![Page 21: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/21.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y
=1
r
![Page 22: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/22.jpg)
• Example:dy
dx= y
• find new coordinates (r , s) to simplify problem
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• obvious choice: r = y and s = x
• change coordinates:
ds
dr=
dx
dy=
1
dy/dx=
1
y=
1
r
![Page 23: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/23.jpg)
Using “magic” coordinates
• change coordinates:
dy
dx= y =⇒ ds
dr=
1
r
• integrate:
s =
∫1
rdr = ln r + C
• change back:x = ln y + C
• solve:y = Cex
![Page 24: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/24.jpg)
Using “magic” coordinates
• change coordinates:
dy
dx= y =⇒ ds
dr=
1
r
• integrate:
s =
∫1
rdr = ln r + C
• change back:x = ln y + C
• solve:y = Cex
![Page 25: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/25.jpg)
Using “magic” coordinates
• change coordinates:
dy
dx= y =⇒ ds
dr=
1
r
• integrate:
s =
∫1
rdr = ln r + C
• change back:x = ln y + C
• solve:y = Cex
![Page 26: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/26.jpg)
Using “magic” coordinates
• change coordinates:
dy
dx= y =⇒ ds
dr=
1
r
• integrate:
s =
∫1
rdr = ln r + C
• change back:x = ln y + C
• solve:y = Cex
![Page 27: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/27.jpg)
Using “magic” coordinates
• change coordinates:
dy
dx= y =⇒ ds
dr=
1
r
• integrate:
s =
∫1
rdr = ln r + C
• change back:x = ln y + C
• solve:y = Cex
![Page 28: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/28.jpg)
Another example of “magic” coordinates
• Example:dy
dx=
y
x+
y2
x3
• a not-so-obvious choice: r =y
xand s = −1
x• change coordinates:
ds
dr=
d(− 1x )
d( yx )= . . . =
1
r2
0 1 2 3 4
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 29: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/29.jpg)
Another example of “magic” coordinates
• Example:dy
dx=
y
x+
y2
x3
• a not-so-obvious choice: r =y
xand s = −1
x• change coordinates:
ds
dr=
d(− 1x )
d( yx )= . . . =
1
r2
0 1 2 3 4
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 30: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/30.jpg)
Another example of “magic” coordinates
• Example:dy
dx=
y
x+
y2
x3
• a not-so-obvious choice: r =y
xand s = −1
x• change coordinates:
ds
dr=
d(− 1x )
d( yx )= . . . =
1
r2
0 1 2 3 4
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 31: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/31.jpg)
Another example of “magic” coordinates
• Example:dy
dx=
y
x+
y2
x3
• a not-so-obvious choice: r =y
xand s = −1
x
• change coordinates:
ds
dr=
d(− 1x )
d( yx )= . . . =
1
r2
0 1 2 3 4
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 32: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/32.jpg)
Another example of “magic” coordinates
• Example:dy
dx=
y
x+
y2
x3
• a not-so-obvious choice: r =y
xand s = −1
x• change coordinates:
ds
dr=
d(− 1x )
d( yx )= . . . =
1
r2
0 1 2 3 4
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 33: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/33.jpg)
Another example of “magic” coordinates
• Example:dy
dx=
y
x+
y2
x3
• a not-so-obvious choice: r =y
xand s = −1
x• change coordinates:
ds
dr=
d(− 1x )
d( yx )= . . . =
1
r2
0 1 2 3 4
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-4
-3
-2
-1
0
r
s
![Page 34: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/34.jpg)
• change coordinates:
dy
dx=
y
x+
y2
x3=⇒ ds
dr=
1
r2
• integrate:
s =
∫1
r2dr = −1
r+ C
• change back:
−1
x= −x
y+ C
• solve:
y =x2
1 + Cx
• Question: how do we come up with r =y
xand s = −1
x?
• Answer: symmetry!
![Page 35: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/35.jpg)
• change coordinates:
dy
dx=
y
x+
y2
x3=⇒ ds
dr=
1
r2
• integrate:
s =
∫1
r2dr = −1
r+ C
• change back:
−1
x= −x
y+ C
• solve:
y =x2
1 + Cx
• Question: how do we come up with r =y
xand s = −1
x?
• Answer: symmetry!
![Page 36: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/36.jpg)
• change coordinates:
dy
dx=
y
x+
y2
x3=⇒ ds
dr=
1
r2
• integrate:
s =
∫1
r2dr = −1
r+ C
• change back:
−1
x= −x
y+ C
• solve:
y =x2
1 + Cx
• Question: how do we come up with r =y
xand s = −1
x?
• Answer: symmetry!
![Page 37: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/37.jpg)
• change coordinates:
dy
dx=
y
x+
y2
x3=⇒ ds
dr=
1
r2
• integrate:
s =
∫1
r2dr = −1
r+ C
• change back:
−1
x= −x
y+ C
• solve:
y =x2
1 + Cx
• Question: how do we come up with r =y
xand s = −1
x?
• Answer: symmetry!
![Page 38: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/38.jpg)
• change coordinates:
dy
dx=
y
x+
y2
x3=⇒ ds
dr=
1
r2
• integrate:
s =
∫1
r2dr = −1
r+ C
• change back:
−1
x= −x
y+ C
• solve:
y =x2
1 + Cx
• Question: how do we come up with r =y
xand s = −1
x?
• Answer: symmetry!
![Page 39: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/39.jpg)
• change coordinates:
dy
dx=
y
x+
y2
x3=⇒ ds
dr=
1
r2
• integrate:
s =
∫1
r2dr = −1
r+ C
• change back:
−1
x= −x
y+ C
• solve:
y =x2
1 + Cx
• Question: how do we come up with r =y
xand s = −1
x?
• Answer: symmetry!
![Page 40: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/40.jpg)
Transforming the plane
• define a mapping T of the plane to itself
• Example: reflection across the y -axis:
• denote this
(x , y) = T (x , y) = (−x , y)
![Page 41: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/41.jpg)
Transforming the plane
• define a mapping T of the plane to itself
• Example: reflection across the y -axis:
• denote this
(x , y) = T (x , y) = (−x , y)
![Page 42: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/42.jpg)
Transforming the plane
• define a mapping T of the plane to itself
• Example: reflection across the y -axis:
• denote this
(x , y) = T (x , y) = (−x , y)
![Page 43: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/43.jpg)
Transforming the plane
• define a mapping T of the plane to itself
• Example: reflection across the y -axis:
Hx,yL
x
y
• denote this
(x , y) = T (x , y) = (−x , y)
![Page 44: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/44.jpg)
Transforming the plane
• define a mapping T of the plane to itself
• Example: reflection across the y -axis:
Hx,yLHx` ,y`L
x
y
• denote this
(x , y) = T (x , y) = (−x , y)
![Page 45: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/45.jpg)
Transforming the plane
• define a mapping T of the plane to itself
• Example: reflection across the y -axis:
Hx,yLHx` ,y`L
x
y
• denote this
(x , y) = T (x , y) = (−x , y)
![Page 46: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/46.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 47: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/47.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 48: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/48.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 49: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/49.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 50: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/50.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 51: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/51.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 52: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/52.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 53: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/53.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 54: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/54.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)
=(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 55: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/55.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 56: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/56.jpg)
Transformation flows
• define a one-parameter family of mappings Tε that maps theplane to itself for each value of the parameter ε
• Examples: Demo
(x , y) = Tε(x , y) = (x + ε, y)
(x , y) = (x , y + ε)
(x , y) = (eεx , y)
(x , y) = (eεx , eεy)
(x , y) = (eεx , e−εy)
(x , y) =( x
1− εx,
y
1− εx
)=
(x , y)
1− εx
(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
![Page 57: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/57.jpg)
Lie group structure
• each of these transformation flows has certain algebraicproperties:
T0 = Id
Tε ◦ Tδ = Tε+δ
T−1ε = T−ε
• so each of these is a group under composition
• each flow is also “nice” as a function of ε
• so each is a one-parameter group of transformations that is“nice” as a function of ε
• these are called one-parameter Lie groups
• now ready to define symmetries of geometric objects
![Page 58: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/58.jpg)
Lie group structure
• each of these transformation flows has certain algebraicproperties:
T0 = Id
Tε ◦ Tδ = Tε+δ
T−1ε = T−ε
• so each of these is a group under composition
• each flow is also “nice” as a function of ε
• so each is a one-parameter group of transformations that is“nice” as a function of ε
• these are called one-parameter Lie groups
• now ready to define symmetries of geometric objects
![Page 59: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/59.jpg)
Lie group structure
• each of these transformation flows has certain algebraicproperties:
T0 = Id
Tε ◦ Tδ = Tε+δ
T−1ε = T−ε
• so each of these is a group under composition
• each flow is also “nice” as a function of ε
• so each is a one-parameter group of transformations that is“nice” as a function of ε
• these are called one-parameter Lie groups
• now ready to define symmetries of geometric objects
![Page 60: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/60.jpg)
Lie group structure
• each of these transformation flows has certain algebraicproperties:
T0 = Id
Tε ◦ Tδ = Tε+δ
T−1ε = T−ε
• so each of these is a group under composition
• each flow is also “nice” as a function of ε
• so each is a one-parameter group of transformations that is“nice” as a function of ε
• these are called one-parameter Lie groups
• now ready to define symmetries of geometric objects
![Page 61: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/61.jpg)
Lie group structure
• each of these transformation flows has certain algebraicproperties:
T0 = Id
Tε ◦ Tδ = Tε+δ
T−1ε = T−ε
• so each of these is a group under composition
• each flow is also “nice” as a function of ε
• so each is a one-parameter group of transformations that is“nice” as a function of ε
• these are called one-parameter Lie groups
• now ready to define symmetries of geometric objects
![Page 62: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/62.jpg)
Lie group structure
• each of these transformation flows has certain algebraicproperties:
T0 = Id
Tε ◦ Tδ = Tε+δ
T−1ε = T−ε
• so each of these is a group under composition
• each flow is also “nice” as a function of ε
• so each is a one-parameter group of transformations that is“nice” as a function of ε
• these are called one-parameter Lie groups
• now ready to define symmetries of geometric objects
![Page 63: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/63.jpg)
Symmetries of a curve
• look at the effect of a transformation flow on a geometricobject
• Example: What happens to the unit circle under each of theprevious transformation flows? Demo
• under most of them, the circle is not mapped to itself• the circle is mapped to itself for rotation through any
angle ε• the circle has a symmetry for each angle ε so the circle
has a one-parameter symmetry flow (in this case, aone-parameter Lie symmetry)
• in general, a geometric object in the plane has a symmetryflow (or a Lie symmetry) if there is a “nice” transformationflow of the plane that maps that object to itself
![Page 64: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/64.jpg)
Symmetries of a curve
• look at the effect of a transformation flow on a geometricobject
• Example: What happens to the unit circle under each of theprevious transformation flows? Demo
• under most of them, the circle is not mapped to itself• the circle is mapped to itself for rotation through any
angle ε• the circle has a symmetry for each angle ε so the circle
has a one-parameter symmetry flow (in this case, aone-parameter Lie symmetry)
• in general, a geometric object in the plane has a symmetryflow (or a Lie symmetry) if there is a “nice” transformationflow of the plane that maps that object to itself
![Page 65: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/65.jpg)
Symmetries of a curve
• look at the effect of a transformation flow on a geometricobject
• Example: What happens to the unit circle under each of theprevious transformation flows? Demo
• under most of them, the circle is not mapped to itself
• the circle is mapped to itself for rotation through anyangle ε
• the circle has a symmetry for each angle ε so the circlehas a one-parameter symmetry flow (in this case, aone-parameter Lie symmetry)
• in general, a geometric object in the plane has a symmetryflow (or a Lie symmetry) if there is a “nice” transformationflow of the plane that maps that object to itself
![Page 66: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/66.jpg)
Symmetries of a curve
• look at the effect of a transformation flow on a geometricobject
• Example: What happens to the unit circle under each of theprevious transformation flows? Demo
• under most of them, the circle is not mapped to itself• the circle is mapped to itself for rotation through any
angle ε
• the circle has a symmetry for each angle ε so the circlehas a one-parameter symmetry flow (in this case, aone-parameter Lie symmetry)
• in general, a geometric object in the plane has a symmetryflow (or a Lie symmetry) if there is a “nice” transformationflow of the plane that maps that object to itself
![Page 67: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/67.jpg)
Symmetries of a curve
• look at the effect of a transformation flow on a geometricobject
• Example: What happens to the unit circle under each of theprevious transformation flows? Demo
• under most of them, the circle is not mapped to itself• the circle is mapped to itself for rotation through any
angle ε• the circle has a symmetry for each angle ε so the circle
has a one-parameter symmetry flow (in this case, aone-parameter Lie symmetry)
• in general, a geometric object in the plane has a symmetryflow (or a Lie symmetry) if there is a “nice” transformationflow of the plane that maps that object to itself
![Page 68: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/68.jpg)
Symmetries of a curve
• look at the effect of a transformation flow on a geometricobject
• Example: What happens to the unit circle under each of theprevious transformation flows? Demo
• under most of them, the circle is not mapped to itself• the circle is mapped to itself for rotation through any
angle ε• the circle has a symmetry for each angle ε so the circle
has a one-parameter symmetry flow (in this case, aone-parameter Lie symmetry)
• in general, a geometric object in the plane has a symmetryflow (or a Lie symmetry) if there is a “nice” transformationflow of the plane that maps that object to itself
![Page 69: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/69.jpg)
First-order differential equations as geometric objects
• geometric view of a first-order ODE as a slope field.
• Examples:
dy
dx= y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
dy
dx=
y
x+
y2
x3
0 1 2 3 4
-2
-1
0
1
2
x
y
• to understand how a slope field transforms, first look at howslopes transform
![Page 70: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/70.jpg)
First-order differential equations as geometric objects
• geometric view of a first-order ODE as a slope field.
• Examples:
dy
dx= y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
dy
dx=
y
x+
y2
x3
0 1 2 3 4
-2
-1
0
1
2
x
y
• to understand how a slope field transforms, first look at howslopes transform
![Page 71: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/71.jpg)
First-order differential equations as geometric objects
• geometric view of a first-order ODE as a slope field.
• Examples:
dy
dx= y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
dy
dx=
y
x+
y2
x3
0 1 2 3 4
-2
-1
0
1
2
x
y
• to understand how a slope field transforms, first look at howslopes transform
![Page 72: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/72.jpg)
First-order differential equations as geometric objects
• geometric view of a first-order ODE as a slope field.
• Examples:
dy
dx= y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
dy
dx=
y
x+
y2
x3
0 1 2 3 4
-2
-1
0
1
2
x
y
• to understand how a slope field transforms, first look at howslopes transform
![Page 73: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/73.jpg)
First-order differential equations as geometric objects
• geometric view of a first-order ODE as a slope field.
• Examples:
dy
dx= y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
dy
dx=
y
x+
y2
x3
0 1 2 3 4
-2
-1
0
1
2
x
y
• to understand how a slope field transforms, first look at howslopes transform
![Page 74: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/74.jpg)
First-order differential equations as geometric objects
• geometric view of a first-order ODE as a slope field.
• Examples:
dy
dx= y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
dy
dx=
y
x+
y2
x3
0 1 2 3 4
-2
-1
0
1
2
x
y
• to understand how a slope field transforms, first look at howslopes transform
![Page 75: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/75.jpg)
Effect of a transformation on slopes
• Example: translation in x
• each point is mapped by (x , y) = (x + ε, y)• each tangent line segment at (x , y) is mapped to a
tangent line segment at (x , y)
• each transformed tangent line segment has slope m thatis the same as the original slope m so
(x , y , m) = Tε(x , y ,m) = (x + ε, y ,m)
![Page 76: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/76.jpg)
Effect of a transformation on slopes
• Example: translation in x
• each point is mapped by (x , y) = (x + ε, y)• each tangent line segment at (x , y) is mapped to a
tangent line segment at (x , y)
• each transformed tangent line segment has slope m thatis the same as the original slope m so
(x , y , m) = Tε(x , y ,m) = (x + ε, y ,m)
![Page 77: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/77.jpg)
Effect of a transformation on slopes
• Example: translation in x
• each point is mapped by (x , y) = (x + ε, y)
• each tangent line segment at (x , y) is mapped to atangent line segment at (x , y)
• each transformed tangent line segment has slope m thatis the same as the original slope m so
(x , y , m) = Tε(x , y ,m) = (x + ε, y ,m)
![Page 78: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/78.jpg)
Effect of a transformation on slopes
• Example: translation in x
• each point is mapped by (x , y) = (x + ε, y)• each tangent line segment at (x , y) is mapped to a
tangent line segment at (x , y)
• each transformed tangent line segment has slope m thatis the same as the original slope m so
(x , y , m) = Tε(x , y ,m) = (x + ε, y ,m)
![Page 79: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/79.jpg)
Effect of a transformation on slopes
• Example: translation in x
• each point is mapped by (x , y) = (x + ε, y)• each tangent line segment at (x , y) is mapped to a
tangent line segment at (x , y)
Hx,yLm
x
y
• each transformed tangent line segment has slope m thatis the same as the original slope m so
(x , y , m) = Tε(x , y ,m) = (x + ε, y ,m)
![Page 80: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/80.jpg)
Effect of a transformation on slopes
• Example: translation in x
• each point is mapped by (x , y) = (x + ε, y)• each tangent line segment at (x , y) is mapped to a
tangent line segment at (x , y)
Hx,yLm
Hx` ,y`L
m`
x
y
• each transformed tangent line segment has slope m thatis the same as the original slope m so
(x , y , m) = Tε(x , y ,m) = (x + ε, y ,m)
![Page 81: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/81.jpg)
Effect of a transformation on slopes
• Example: translation in x
• each point is mapped by (x , y) = (x + ε, y)• each tangent line segment at (x , y) is mapped to a
tangent line segment at (x , y)
Hx,yLm
Hx` ,y`L
m`
x
y
• each transformed tangent line segment has slope m thatis the same as the original slope m so
(x , y , m) = Tε(x , y ,m) = (x + ε, y ,m)
![Page 82: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/82.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 83: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/83.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 84: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/84.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 85: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/85.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 86: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/86.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 87: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/87.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun
=rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 88: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/88.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun
= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 89: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/89.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run
= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 90: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/90.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 91: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/91.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 92: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/92.jpg)
• Example: scaling in x : (x , y) = T (x , y) = (eεx , y)
• tangent line segments will be scaled in the x-direction
Hx,yLm
Hx` ,y`L
m`
x
y
• scaling in x will scale run and have no effect on rise so
m =ˆrise
ˆrun=
rise
eεrun= e−ε rise
run= e−εm
• so (x , y , m) = Tε(x , y ,m) = (eεx , y , e−εm)
Demo
![Page 93: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/93.jpg)
Symmetries of a first-order differential equation
• start with a differential equationdy
dx= f (x , y).
• look at the effect of a transformation T on a slope field
• Example:dy
dx= y under reflection across the x-axis
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x`
y`
• T is a symmetry of the differential equation if the slope fieldmaps to itself (so each solution is mapped to a solution)
![Page 94: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/94.jpg)
Symmetries of a first-order differential equation
• start with a differential equationdy
dx= f (x , y).
• look at the effect of a transformation T on a slope field
• Example:dy
dx= y under reflection across the x-axis
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x`
y`
• T is a symmetry of the differential equation if the slope fieldmaps to itself (so each solution is mapped to a solution)
![Page 95: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/95.jpg)
Symmetries of a first-order differential equation
• start with a differential equationdy
dx= f (x , y).
• look at the effect of a transformation T on a slope field
• Example:dy
dx= y under reflection across the x-axis
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x`
y`
• T is a symmetry of the differential equation if the slope fieldmaps to itself (so each solution is mapped to a solution)
![Page 96: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/96.jpg)
Symmetries of a first-order differential equation
• start with a differential equationdy
dx= f (x , y).
• look at the effect of a transformation T on a slope field
• Example:dy
dx= y under reflection across the x-axis
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x`
y`
• T is a symmetry of the differential equation if the slope fieldmaps to itself (so each solution is mapped to a solution)
![Page 97: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/97.jpg)
Symmetries of a first-order differential equation
• start with a differential equationdy
dx= f (x , y).
• look at the effect of a transformation T on a slope field
• Example:dy
dx= y under reflection across the x-axis
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x`
y`
• T is a symmetry of the differential equation if the slope fieldmaps to itself (so each solution is mapped to a solution)
![Page 98: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/98.jpg)
Symmetries of a first-order differential equation
• start with a differential equationdy
dx= f (x , y).
• look at the effect of a transformation T on a slope field
• Example:dy
dx= y under reflection across the x-axis
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x`
y`
• T is a symmetry of the differential equation if the slope fieldmaps to itself (so each solution is mapped to a solution)
![Page 99: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/99.jpg)
Symmetries of a first-order differential equation
• start with a differential equationdy
dx= f (x , y).
• look at the effect of a transformation T on a slope field
• Example:dy
dx= y under reflection across the x-axis
-2 -1 0 1 2
-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x`
y`
• T is a symmetry of the differential equation if the slope fieldmaps to itself (so each solution is mapped to a solution)
![Page 100: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/100.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x :
a symmetry flow
• scaling in x :
not a symmetry flow
• translation in y :
not a symmetry flow
• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 101: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/101.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x :
a symmetry flow
• scaling in x :
not a symmetry flow
• translation in y :
not a symmetry flow
• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 102: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/102.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x :
a symmetry flow• scaling in x :
not a symmetry flow
• translation in y :
not a symmetry flow
• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 103: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/103.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x :
a symmetry flow
• scaling in x :
not a symmetry flow• translation in y :
not a symmetry flow
• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 104: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/104.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x :
a symmetry flow
• scaling in x :
not a symmetry flow
• translation in y :
not a symmetry flow• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 105: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/105.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x :
a symmetry flow
• scaling in x :
not a symmetry flow
• translation in y :
not a symmetry flow
• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 106: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/106.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x :
not a symmetry flow
• translation in y :
not a symmetry flow
• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 107: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/107.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y :
not a symmetry flow
• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 108: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/108.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y :
a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 109: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/109.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y : a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 110: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/110.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y : a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 111: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/111.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y : a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 112: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/112.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y : a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y :
not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 113: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/113.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y : a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y : not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 114: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/114.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y : a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y : not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 115: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/115.jpg)
• our interest is in symmetry under a transformation flow
• Example: exploredy
dx= y under various transformation flows
Demo
• translation in x : a symmetry flow• scaling in x : not a symmetry flow• translation in y : not a symmetry flow• scaling in y : a symmetry flow
• Example: exploredy
dx=
y
x+
y2
x3under various transformation
flows Demo
• scaling in y : not a symmetry flow
• projective transformation Tε(x , y) =(x , y)
1− εx:
a symmetryflow
• before working with symmetries of a differential equation, lookat a convenient way to picture a transformation flow
![Page 116: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/116.jpg)
Visualizing a transformation flow
• have looked at how a grid of points moves
• can also look at paths traced out by points
• Example: (x , y) = (eεx , e−εy)
-2 -1 0 1 2-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
• denote the tangent vector field ~X = (ξ, η)
![Page 117: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/117.jpg)
Visualizing a transformation flow
• have looked at how a grid of points moves
• can also look at paths traced out by points
• Example: (x , y) = (eεx , e−εy)
-2 -1 0 1 2-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
• denote the tangent vector field ~X = (ξ, η)
![Page 118: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/118.jpg)
Visualizing a transformation flow
• have looked at how a grid of points moves
• can also look at paths traced out by points
• Example: (x , y) = (eεx , e−εy)
-2 -1 0 1 2-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
• denote the tangent vector field ~X = (ξ, η)
![Page 119: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/119.jpg)
Visualizing a transformation flow
• have looked at how a grid of points moves
• can also look at paths traced out by points
• Example: (x , y) = (eεx , e−εy)
-2 -1 0 1 2-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
• denote the tangent vector field ~X = (ξ, η)
![Page 120: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/120.jpg)
Visualizing a transformation flow
• have looked at how a grid of points moves
• can also look at paths traced out by points
• Example: (x , y) = (eεx , e−εy)
-2 -1 0 1 2-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
• denote the tangent vector field ~X = (ξ, η)
![Page 121: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/121.jpg)
Visualizing a transformation flow
• have looked at how a grid of points moves
• can also look at paths traced out by points
• Example: (x , y) = (eεx , e−εy)
-2 -1 0 1 2-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
• denote the tangent vector field ~X = (ξ, η)
![Page 122: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/122.jpg)
Visualizing a transformation flow
• have looked at how a grid of points moves
• can also look at paths traced out by points
• Example: (x , y) = (eεx , e−εy)
-2 -1 0 1 2-2
-1
0
1
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
x
y
• denote the tangent vector field ~X = (ξ, η)
![Page 123: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/123.jpg)
Computing the tangent vector field
• the tangent vector field is given by
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
• Example: (x , y) = (eεx , e−εy)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=(eεx ,−e−εy
)∣∣∣ε=0
= (x ,−y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 124: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/124.jpg)
Computing the tangent vector field
• the tangent vector field is given by
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
• Example: (x , y) = (eεx , e−εy)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=(eεx ,−e−εy
)∣∣∣ε=0
= (x ,−y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 125: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/125.jpg)
Computing the tangent vector field
• the tangent vector field is given by
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
• Example: (x , y) = (eεx , e−εy)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=(eεx ,−e−εy
)∣∣∣ε=0
= (x ,−y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 126: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/126.jpg)
Computing the tangent vector field
• the tangent vector field is given by
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
• Example: (x , y) = (eεx , e−εy)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=(eεx ,−e−εy
)∣∣∣ε=0
= (x ,−y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 127: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/127.jpg)
Computing the tangent vector field
• the tangent vector field is given by
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
• Example: (x , y) = (eεx , e−εy)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=(eεx ,−e−εy
)∣∣∣ε=0
= (x ,−y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 128: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/128.jpg)
Computing the tangent vector field
• the tangent vector field is given by
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
• Example: (x , y) = (eεx , e−εy)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=(eεx ,−e−εy
)∣∣∣ε=0
= (x ,−y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 129: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/129.jpg)
Computing the tangent vector field
• the tangent vector field is given by
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
• Example: (x , y) = (eεx , e−εy)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=(eεx ,−e−εy
)∣∣∣ε=0
= (x ,−y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 130: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/130.jpg)
• Example: (x , y) =( x
1− εx,
y
1− εx
)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=
(x2
(1− εx)2,
xy
(1− εx)2
)∣∣∣∣∣ε=0
=(x2, xy
)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 131: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/131.jpg)
• Example: (x , y) =( x
1− εx,
y
1− εx
)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=
(x2
(1− εx)2,
xy
(1− εx)2
)∣∣∣∣∣ε=0
=(x2, xy
)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 132: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/132.jpg)
• Example: (x , y) =( x
1− εx,
y
1− εx
)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=
(x2
(1− εx)2,
xy
(1− εx)2
)∣∣∣∣∣ε=0
=(x2, xy
)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 133: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/133.jpg)
• Example: (x , y) =( x
1− εx,
y
1− εx
)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=
(x2
(1− εx)2,
xy
(1− εx)2
)∣∣∣∣∣ε=0
=(x2, xy
)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 134: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/134.jpg)
• Example: (x , y) =( x
1− εx,
y
1− εx
)
~X =
(dx
dε,
dy
dε
)∣∣∣∣∣ε=0
=
(x2
(1− εx)2,
xy
(1− εx)2
)∣∣∣∣∣ε=0
=(x2, xy
)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 135: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/135.jpg)
• a few more examples of tangent vector fields:
Scaling in x(x , y) = (eεx , y)
~X = (x , 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
Rotation(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
~X = (−y , x)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 136: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/136.jpg)
• a few more examples of tangent vector fields:
Scaling in x(x , y) = (eεx , y)
~X = (x , 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
Rotation(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
~X = (−y , x)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 137: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/137.jpg)
• a few more examples of tangent vector fields:
Scaling in x(x , y) = (eεx , y)
~X = (x , 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
Rotation(x , y) = (x cos ε− y sin ε, x sin ε+ y cos ε)
~X = (−y , x)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
![Page 138: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/138.jpg)
Using a symmetry to solve the differential equation
• find coordinates (r , s) in which the symmetry field is verticaland uniform
~X = (x2, xy)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• in the new coordinates, differential equation reduces to anantiderivative problem since symmetry maps solutions tosolutions by translation in the dependent variable
![Page 139: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/139.jpg)
Using a symmetry to solve the differential equation
• find coordinates (r , s) in which the symmetry field is verticaland uniform
~X = (x2, xy)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• in the new coordinates, differential equation reduces to anantiderivative problem since symmetry maps solutions tosolutions by translation in the dependent variable
![Page 140: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/140.jpg)
Using a symmetry to solve the differential equation
• find coordinates (r , s) in which the symmetry field is verticaland uniform
~X = (x2, xy)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• in the new coordinates, differential equation reduces to anantiderivative problem since symmetry maps solutions tosolutions by translation in the dependent variable
![Page 141: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/141.jpg)
Using a symmetry to solve the differential equation
• find coordinates (r , s) in which the symmetry field is verticaland uniform
~X = (x2, xy)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• in the new coordinates, differential equation reduces to anantiderivative problem since symmetry maps solutions tosolutions by translation in the dependent variable
![Page 142: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/142.jpg)
Using a symmetry to solve the differential equation
• find coordinates (r , s) in which the symmetry field is verticaland uniform
~X = (x2, xy)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• in the new coordinates, differential equation reduces to anantiderivative problem since symmetry maps solutions tosolutions by translation in the dependent variable
![Page 143: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/143.jpg)
• Example: translation in x: ~X = (1, 0)
• from the geometry, can see that choosing r = y , s = x works
~X = (1, 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is a symmetry flow fordy
dx= y
• in these “magic coordinates”, this ODE becomesds
dr=
1
r
![Page 144: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/144.jpg)
• Example: translation in x: ~X = (1, 0)
• from the geometry, can see that choosing r = y , s = x works
~X = (1, 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is a symmetry flow fordy
dx= y
• in these “magic coordinates”, this ODE becomesds
dr=
1
r
![Page 145: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/145.jpg)
• Example: translation in x: ~X = (1, 0)
• from the geometry, can see that choosing r = y , s = x works
~X = (1, 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is a symmetry flow fordy
dx= y
• in these “magic coordinates”, this ODE becomesds
dr=
1
r
![Page 146: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/146.jpg)
• Example: translation in x: ~X = (1, 0)
• from the geometry, can see that choosing r = y , s = x works
~X = (1, 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is a symmetry flow fordy
dx= y
• in these “magic coordinates”, this ODE becomesds
dr=
1
r
![Page 147: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/147.jpg)
• Example: translation in x: ~X = (1, 0)
• from the geometry, can see that choosing r = y , s = x works
~X = (1, 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is a symmetry flow fordy
dx= y
• in these “magic coordinates”, this ODE becomesds
dr=
1
r
![Page 148: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/148.jpg)
• Example: translation in x: ~X = (1, 0)
• from the geometry, can see that choosing r = y , s = x works
~X = (1, 0)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is a symmetry flow fordy
dx= y
• in these “magic coordinates”, this ODE becomesds
dr=
1
r
![Page 149: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/149.jpg)
• Example: scaling in y : ~X = (0, y)
• from the geometry, see that r = x and s is some function ofy ; a choice that works is s = ln y
~X = (0, y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is also a symmetry flow fordy
dx= y so can now transform
the differential equation to the new coordinates
![Page 150: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/150.jpg)
• Example: scaling in y : ~X = (0, y)
• from the geometry, see that r = x and s is some function ofy ; a choice that works is s = ln y
~X = (0, y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is also a symmetry flow fordy
dx= y so can now transform
the differential equation to the new coordinates
![Page 151: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/151.jpg)
• Example: scaling in y : ~X = (0, y)
• from the geometry, see that r = x and s is some function ofy ; a choice that works is s = ln y
~X = (0, y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is also a symmetry flow fordy
dx= y so can now transform
the differential equation to the new coordinates
![Page 152: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/152.jpg)
• Example: scaling in y : ~X = (0, y)
• from the geometry, see that r = x and s is some function ofy ; a choice that works is s = ln y
~X = (0, y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is also a symmetry flow fordy
dx= y so can now transform
the differential equation to the new coordinates
![Page 153: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/153.jpg)
• Example: scaling in y : ~X = (0, y)
• from the geometry, see that r = x and s is some function ofy ; a choice that works is s = ln y
~X = (0, y)
-2 -1 0 1 2
-2
-1
0
1
2
x
y
~X = (0, 1)
-2 -1 0 1 2
-2
-1
0
1
2
r
s
• this is also a symmetry flow fordy
dx= y so can now transform
the differential equation to the new coordinates
![Page 154: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/154.jpg)
Finding “magic” coordinates
• choose r(x , y) so that r = constant curves are tangent to~X = (ξ, η)
• Example: ~X = (x2, xy)
• equivalent to choosing r(x , y) so that the derivative in thedirection of ~X is 0:
~X · ~∇r = ξ∂r
∂x+ η
∂r
∂y= 0
![Page 155: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/155.jpg)
Finding “magic” coordinates
• choose r(x , y) so that r = constant curves are tangent to~X = (ξ, η)
• Example: ~X = (x2, xy)
• equivalent to choosing r(x , y) so that the derivative in thedirection of ~X is 0:
~X · ~∇r = ξ∂r
∂x+ η
∂r
∂y= 0
![Page 156: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/156.jpg)
Finding “magic” coordinates
• choose r(x , y) so that r = constant curves are tangent to~X = (ξ, η)
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• equivalent to choosing r(x , y) so that the derivative in thedirection of ~X is 0:
~X · ~∇r = ξ∂r
∂x+ η
∂r
∂y= 0
![Page 157: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/157.jpg)
Finding “magic” coordinates
• choose r(x , y) so that r = constant curves are tangent to~X = (ξ, η)
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• equivalent to choosing r(x , y) so that the derivative in thedirection of ~X is 0:
~X · ~∇r = ξ∂r
∂x+ η
∂r
∂y= 0
![Page 158: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/158.jpg)
Finding “magic” coordinates
• choose r(x , y) so that r = constant curves are tangent to~X = (ξ, η)
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• equivalent to choosing r(x , y) so that the derivative in thedirection of ~X is 0:
~X · ~∇r = ξ∂r
∂x+ η
∂r
∂y= 0
![Page 159: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/159.jpg)
Finding “magic” coordinates
• choose r(x , y) so that r = constant curves are tangent to~X = (ξ, η)
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• equivalent to choosing r(x , y) so that the derivative in thedirection of ~X is 0:
~X · ~∇r = ξ∂r
∂x+ η
∂r
∂y= 0
![Page 160: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/160.jpg)
• choose s(x , y) so that s = constant curves are nowheretangent to r = constant curves and the derivative in thedirection of ~X is uniform:
~X · ~∇s = ξ∂s
∂x+ η
∂s
∂y= 1
• Example: ~X = (x2, xy)
• looks intimidating but not so bad in practice since need only aspecific solution rather than the general solution
![Page 161: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/161.jpg)
• choose s(x , y) so that s = constant curves are nowheretangent to r = constant curves and the derivative in thedirection of ~X is uniform:
~X · ~∇s = ξ∂s
∂x+ η
∂s
∂y= 1
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• looks intimidating but not so bad in practice since need only aspecific solution rather than the general solution
![Page 162: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/162.jpg)
• choose s(x , y) so that s = constant curves are nowheretangent to r = constant curves and the derivative in thedirection of ~X is uniform:
~X · ~∇s = ξ∂s
∂x+ η
∂s
∂y= 1
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• looks intimidating but not so bad in practice since need only aspecific solution rather than the general solution
![Page 163: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/163.jpg)
• choose s(x , y) so that s = constant curves are nowheretangent to r = constant curves and the derivative in thedirection of ~X is uniform:
~X · ~∇s = ξ∂s
∂x+ η
∂s
∂y= 1
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• looks intimidating but not so bad in practice since need only aspecific solution rather than the general solution
![Page 164: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/164.jpg)
• choose s(x , y) so that s = constant curves are nowheretangent to r = constant curves and the derivative in thedirection of ~X is uniform:
~X · ~∇s = ξ∂s
∂x+ η
∂s
∂y= 1
• Example: ~X = (x2, xy)
0 1 2 3 4
-2
-1
0
1
2
x
y
• looks intimidating but not so bad in practice since need only aspecific solution rather than the general solution
![Page 165: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/165.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 166: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/166.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 167: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/167.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 168: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/168.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy
=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 169: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/169.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y
=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 170: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/170.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant
so r =y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 171: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/171.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 172: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/172.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1
=⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 173: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/173.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2
=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 174: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/174.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 175: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/175.jpg)
• Example: ~X = (x2, xy)
• must find a solution to
x2 ∂r
∂x+ xy
∂r
∂y= 0
x2 ∂s
∂x+ xy
∂s
∂y= 1
• top equation:r is constant along curves given by
dx
x2=
dy
xy=⇒ dx
x=
dy
y=⇒ y
x= constant so r =
y
xworks
• solve bottom equation by looking for s depending only on x so
x2 ∂s
∂x+ xy · 0 = 1 =⇒ ∂s
∂x=
1
x2=⇒ s = −1
x
• these are the “magic” coordinates we used at the start
![Page 176: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/176.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η)
Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 177: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/177.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η)
Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 178: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/178.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η)
Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 179: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/179.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η)
Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 180: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/180.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η)
Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 181: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/181.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η)
Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 182: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/182.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η)
Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 183: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/183.jpg)
Review the general plan
• start with a differential equation in the formdy
dx= f (x , y)
meaning slopes can vary in both x and y
• find a symmetry flow for the differential equation with tangentvector field ~X = (ξ, η) Wait a minute, how do we do that!?
• for that symmetry flow, find new coordinates r and s so thatthe tangent vector field is ~X = (0, 1)
• in the new coordinates, slopes can vary only in r and not in ssince translation in s maps slope field to slope field
• thus, in the new coordinates, the differential equation has the
formds
dr= g(r)
• integrate!
• change back to the original coordinates
![Page 184: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/184.jpg)
Finding the symmetries of a differential equation
• defining condition: (x , y) = Tε(x , y) is a symmetry flow if
dy
dx= f (x , y) =⇒ dy
dx= f (x , y)
• strategy: determine the tangent vector field ~X = (ξ, η) bylinearizing the defining condition
• start with
(x , y) = (x , y) + ε(ξ, η) + higher-order terms to be ignored
• substitute into defining condition:
d(y + ε η)
d(x + ε ξ)= f (x + ε ξ, y + ε η)
• after the dust settles:
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• first-order linear PDE for ξ(x , y) and η(x , y)
![Page 185: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/185.jpg)
Finding the symmetries of a differential equation
• defining condition: (x , y) = Tε(x , y) is a symmetry flow if
dy
dx= f (x , y) =⇒ dy
dx= f (x , y)
• strategy: determine the tangent vector field ~X = (ξ, η) bylinearizing the defining condition
• start with
(x , y) = (x , y) + ε(ξ, η) + higher-order terms to be ignored
• substitute into defining condition:
d(y + ε η)
d(x + ε ξ)= f (x + ε ξ, y + ε η)
• after the dust settles:
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• first-order linear PDE for ξ(x , y) and η(x , y)
![Page 186: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/186.jpg)
Finding the symmetries of a differential equation
• defining condition: (x , y) = Tε(x , y) is a symmetry flow if
dy
dx= f (x , y) =⇒ dy
dx= f (x , y)
• strategy: determine the tangent vector field ~X = (ξ, η) bylinearizing the defining condition
• start with
(x , y) = (x , y) + ε(ξ, η) + higher-order terms to be ignored
• substitute into defining condition:
d(y + ε η)
d(x + ε ξ)= f (x + ε ξ, y + ε η)
• after the dust settles:
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• first-order linear PDE for ξ(x , y) and η(x , y)
![Page 187: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/187.jpg)
Finding the symmetries of a differential equation
• defining condition: (x , y) = Tε(x , y) is a symmetry flow if
dy
dx= f (x , y) =⇒ dy
dx= f (x , y)
• strategy: determine the tangent vector field ~X = (ξ, η) bylinearizing the defining condition
• start with
(x , y) = (x , y) + ε(ξ, η) + higher-order terms to be ignored
• substitute into defining condition:
d(y + ε η)
d(x + ε ξ)= f (x + ε ξ, y + ε η)
• after the dust settles:
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• first-order linear PDE for ξ(x , y) and η(x , y)
![Page 188: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/188.jpg)
Finding the symmetries of a differential equation
• defining condition: (x , y) = Tε(x , y) is a symmetry flow if
dy
dx= f (x , y) =⇒ dy
dx= f (x , y)
• strategy: determine the tangent vector field ~X = (ξ, η) bylinearizing the defining condition
• start with
(x , y) = (x , y) + ε(ξ, η) + higher-order terms to be ignored
• substitute into defining condition:
d(y + ε η)
d(x + ε ξ)= f (x + ε ξ, y + ε η)
• after the dust settles:
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• first-order linear PDE for ξ(x , y) and η(x , y)
![Page 189: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/189.jpg)
Finding the symmetries of a differential equation
• defining condition: (x , y) = Tε(x , y) is a symmetry flow if
dy
dx= f (x , y) =⇒ dy
dx= f (x , y)
• strategy: determine the tangent vector field ~X = (ξ, η) bylinearizing the defining condition
• start with
(x , y) = (x , y) + ε(ξ, η) + higher-order terms to be ignored
• substitute into defining condition:
d(y + ε η)
d(x + ε ξ)= f (x + ε ξ, y + ε η)
• after the dust settles:
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• first-order linear PDE for ξ(x , y) and η(x , y)
![Page 190: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/190.jpg)
Finding the symmetries of a differential equation
• defining condition: (x , y) = Tε(x , y) is a symmetry flow if
dy
dx= f (x , y) =⇒ dy
dx= f (x , y)
• strategy: determine the tangent vector field ~X = (ξ, η) bylinearizing the defining condition
• start with
(x , y) = (x , y) + ε(ξ, η) + higher-order terms to be ignored
• substitute into defining condition:
d(y + ε η)
d(x + ε ξ)= f (x + ε ξ, y + ε η)
• after the dust settles:
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• first-order linear PDE for ξ(x , y) and η(x , y)
![Page 191: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/191.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 192: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/192.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 193: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/193.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ
• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 194: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/194.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 195: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/195.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 196: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/196.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 197: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/197.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 198: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/198.jpg)
∂η
∂x+ f(∂η∂y− ∂ξ
∂x
)− f 2 ∂ξ
∂y=∂f
∂xξ +
∂f
∂yη
• Example:dy
dx= y so f (x , y) = y
• try ξ = ax + by + c and η = αx + βy + γ• substitute:
α + (β − a)y − by2 = αx + βy + γ
• match coefficients:
1 : α = γ
x : 0 = α
y : β − a = β
y2 : b = 0
=⇒ξ = c
η = βy
• so ~X = (1, 0) and ~X = (0, y) are symmetry vector fields
![Page 199: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/199.jpg)
Topics for another time
• Classifying first-order ODEs by symmetry
• Symmetries of higher-order ODEs
• Symmetries of PDEs
• Finding invariant solutions (example: fundamental solution forthe heat equation)
• Algebraic structure of symmetries: Lie groups and Lie algebras
• Variational symmetries
• Nonclassical symmetries
![Page 200: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/200.jpg)
Topics for another time
• Classifying first-order ODEs by symmetry
• Symmetries of higher-order ODEs
• Symmetries of PDEs
• Finding invariant solutions (example: fundamental solution forthe heat equation)
• Algebraic structure of symmetries: Lie groups and Lie algebras
• Variational symmetries
• Nonclassical symmetries
![Page 201: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/201.jpg)
Topics for another time
• Classifying first-order ODEs by symmetry
• Symmetries of higher-order ODEs
• Symmetries of PDEs
• Finding invariant solutions (example: fundamental solution forthe heat equation)
• Algebraic structure of symmetries: Lie groups and Lie algebras
• Variational symmetries
• Nonclassical symmetries
![Page 202: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/202.jpg)
Topics for another time
• Classifying first-order ODEs by symmetry
• Symmetries of higher-order ODEs
• Symmetries of PDEs
• Finding invariant solutions (example: fundamental solution forthe heat equation)
• Algebraic structure of symmetries: Lie groups and Lie algebras
• Variational symmetries
• Nonclassical symmetries
![Page 203: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/203.jpg)
Topics for another time
• Classifying first-order ODEs by symmetry
• Symmetries of higher-order ODEs
• Symmetries of PDEs
• Finding invariant solutions (example: fundamental solution forthe heat equation)
• Algebraic structure of symmetries: Lie groups and Lie algebras
• Variational symmetries
• Nonclassical symmetries
![Page 204: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/204.jpg)
Topics for another time
• Classifying first-order ODEs by symmetry
• Symmetries of higher-order ODEs
• Symmetries of PDEs
• Finding invariant solutions (example: fundamental solution forthe heat equation)
• Algebraic structure of symmetries: Lie groups and Lie algebras
• Variational symmetries
• Nonclassical symmetries
![Page 205: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/205.jpg)
Topics for another time
• Classifying first-order ODEs by symmetry
• Symmetries of higher-order ODEs
• Symmetries of PDEs
• Finding invariant solutions (example: fundamental solution forthe heat equation)
• Algebraic structure of symmetries: Lie groups and Lie algebras
• Variational symmetries
• Nonclassical symmetries
![Page 206: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/206.jpg)
A few references
Peter Hydon, Symmetry Methods for DifferentialEquations: A Beginner’s Guide, Cambridge, 2000.
Peter Olver, Applications of Lie Groups to Differ-ential Equations 2nd ed., Springer, 1993.
![Page 207: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/207.jpg)
A few references
Peter Hydon, Symmetry Methods for DifferentialEquations: A Beginner’s Guide, Cambridge, 2000.
Peter Olver, Applications of Lie Groups to Differ-ential Equations 2nd ed., Springer, 1993.
![Page 208: Using symmetry to solve differential equationsmartinj/research/LewisClark_March2012.pdfUsing symmetry to solve di erential equations Martin Jackson Mathematics and Computer Science,](https://reader034.vdocuments.us/reader034/viewer/2022042206/5ea923b93f208d3cf14cb6be/html5/thumbnails/208.jpg)
A few references
Peter Hydon, Symmetry Methods for DifferentialEquations: A Beginner’s Guide, Cambridge, 2000.
Peter Olver, Applications of Lie Groups to Differ-ential Equations 2nd ed., Springer, 1993.