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in curvature to achieve aesthetic curves [17]. Then, Ling Xu and David Mould proposed tocontinuously change the charge on a simulated particle so that it can trace out a complex curve
with continuously varying curvature. They showed some examples of abstract figures created by
this method and also show how some stylized representational forms, including fire, hair, and trees,can be drawn with magnetic curves [18].
In this paper, we give a new variational approach to studies the magnetic flow asociated with theKilling magnetic field in a D3 Riemannian space ).,( 3 gM And then, we investigate the
trajectories of the magnetic fields called as N-magnetic and B-magnetic curves. Moreover, weobtain some solitions of the Lorentz force eqution and give some examples of these curves withdraw their pictures by using Mathematica.
2. PreliminariesLet M be a )2( n dimensional oriented Riemannian manifold. The Lorentz force of a magnetic
field F on M is defined to be a skew symetric operator given by
(2) ),()),(( YXFYXg for all ).(, MYX
The magnetic trajectories of F are curves on M which satisfy theLorentz equation
(3) ).(
The mixed productof the vector fields )(,, MZYX is defined by
(4) ).,,(),( ZYXdvZYXg g
LetV
be a Killing vector field onM
and gVV dvF
be the corresponding Killing magneticfield, where is denoted the inner product. Then theLorentz forceof the
VF is
(5) .)( XVX
Consequently, theLorentz force equationmay be written as
(6) . V
A unit speed curve is a magnetic trajectoryof a magnetic field V if and only if V can be
written along as
(7) )()()()( sBssTsV where the function )(s associated with each magnetic curve will be called its quasislope
measured with respect to the magnetic field V (see for details [3]).
Proposition 2.1. [3]Let 3: MI R be a curve in a 3D oriented Riemannian Manifold
),( 3 gM and V be a vector field along the curve . One can take a variation of in the
direction of V, say, a map
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3),(: MI which satisfies ),()0,( ss ).(),( sVtss
In this setting, we have the
following functions,
1. the speed function ,),(),( tstsvs
2. the curvature function ),( ts of ),(st
3. the torsion function ),( ts
of ).(s
t
The variations of those functions at ,0t
),,),((),(
),(),),((?
1),()()10(
),,),((),(2),(),()()9(
,),(),()()8(
2
0
2
0
0
BNTVRgTVg
BVBTTVRVgtst
V
NTTVRgTVgNVgtst
V
vTVgtst
vvV
T
T
s
T
t
TT
t
T
t
where R is curvature tensor of .3
M
Proposition 2.2. [3]Let )(sV be the restriction to )(s of a Killing vector field ,say, V of3
M ; then.0)()()( VVvV
Proposition 2.3.[12] Let be a unit speed space curve with 0)( s . Then is a slant helix
if and only if
)()(
2/3
22
2
ss
is a constant function.
3.New Kind of Magnetic Curves in 3D OrientedRiemannian Manifolds
3.1. N-Magnetic Curves. In this section, we defined a new kind of magnetic curve called N-
magnetic curve in oriented 3D Manifolds, .,3 gM Moreover, we obtain some characterizationsand examples of the curve.
Definition 3.1.Let 3: MI R be a curve in D3 oriented Riemannian space ),( 3 gM
and F be a magnetic field on .M We call the curve is a N-magnetic curve if the normal
vector field of the curve satisfy the Lorentz force equation, that is,
(12) .)( NVNN
Proposition3.1.Let be a unit speed N-magnetic curve in D3 oriented Riemannian space
),( 3 gM with the Frenet apparatus ,,,, BNT . Then we have the Serret-Frenet formulae:
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(12) .
00
0
00
B
N
T
B
N
T
T
T
T
And then the Lorentz force in the Frenet frame written as
(13) .
0
0
0
)(
)(
)(
1
1
B
N
T
B
N
T
where1 is a certain function.
Proof. Let be a unit speed N-magnetic curve in D3 oriented Riemannian space ),( 3 gM
with the Frenet apparatus ,,,, BNT . Since },,,{)( BNTspanT we have
BNTT )(
and thus
1)),((
)(),()),(()),((
0)),((
BTg
BTgTNgTNgNTg
TTg
Therefore we can write
BNT 1)(
Similarly, we can easily calculate that
.)(
)(
1 NTB
BTN
These complete the proof.
Proposition 3.2. Let be a unit speed N-magnetic trajectory of a magnetic field V if and
only if V can be written along the curve as
(14) .1
BNTV
Proof. Let be a unit speed N-magnetic trajectory of a magnetic field .V Using the Proposition
3.1 and Eq. (5), we can easily see that
.1 BNTV This completes the proof.
Theorem 3.3.(Main result) Let V be a Killing vector field on a simply connected space form
.),(3 gCM Then, the unit speed N-magnetic trajectories of VgCM ,),(3 are curves withcurvature and torsion satisfying
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(15)
11
2
11 ,01
C
where C is curvature of the Riemannian space 3M and1 satisfy
.1
BNTV
Proof. Let Vbe a magnetic field in a Riemannian 3D manifold. Then V satisfy Eq. (14).
Differentiating Eq. (14) with respect to s , we have
(16) ,)'()( 111 BNTVT
By differentiating of Eq. (24) with respect to s and using the Serret-Frenet formulas, we get
(17) .)'())'(( 111112
BNTVT
Proposition 2.2 implies that ,0)( vV so from Eq. (a) and Eq. (24), we get
(18) 01 then, if Eq. (24) and Eq. (25) are considered with 0)( V in Proposition 2.2, we obtain
.0),),(()'( 11
NTTVRg
In particular, if 3M has constant curvature ,C then 0),(),),(( NVCgNTTVRg and so,
(19) .0)'( 11
Similarly, if we combine Eq. (24) and Eq. (25) wih 0)( V in Proposition 2.2, we have
.0),),((),),(()'(' 12
11 BTTVRBNTVRg
Hence, if 3M has constant curvature ,C then CBVCgBTTVRg ),(),),(( and
0),),(( BNTVRg give us
(20) .0)'(' 12
11
C
Finally, considering Eq. (26) and Eq. (27) with Eq. (28), this implies
.,01
11
2
11
C
This completes the proof.
Corollary 3.4. Considering1 is a non-zero constant function, we can easily see that the N-
magnetic curve is a curve in the Euclidean 3 space.
Proof. Using the similar method of the above proof, it is obvious.
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Corollary 3.5. Let be a unit speed N-magnetic curve in D3 oriented Riemannian manifold
gCM ),(3 . If the function1 non-zero constant, then the curve is a slant helix. Moreover,
the axis of the slant helix is the the vector field .V
Proof. We assume that is a N-magnetic curve in Euclidean 3-space with non-zero constant
function 1 , then from Eq. (15), we get
(21)1
'
which implies that
constant22 Also, Eq. (17) carry out the following equation with the different point of view, we get
constant' 221
or
.122
2
where1 and
22 are constant functions. By the Proposition 2.3 we obtain that is a
slant helix in Euclidean 3-space. These complete the proof.
Example 3.1.We consider a N-magnetic curve in Euclidean 3 space is defined by
15
)3cos(4),
64
2sin
64
9sin(
5
8),
64
2cos
64
9cos(
5
8)(
sssss
s
The picture of the N-magnetic curve is rendered in Figure 1. The curve has the following
curvature and torsion given by
ss
ss
3sin4)(
3cos4)(
and using the corollary 3.5 we can easily see that .11 So is a N-magnetic curve. The
picture of the curve is rendered in Figure 1.
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)1(curvemagnetic-N1.Figure 1
Corollary 3.6.Let be a N-magnetic curve in Euclidean 3-space with1 is zero, then
is a circular helix. Moreover, the axis of the circular helix is the vector field .V
Proof.It is obvious from Eq. (15).
Example 3.2.We consider a N-magnetic curve in Euclidean 3 space is defined by
.2
,2
sin,2
cos)(
ssss
The curve has the following curvature and torsion given by
2
1)(
21)(
s
s
and using the corollary 3.5 we can easily see that .01 So is a N-magnetic curve.The
picture of the curve is rendered in Figure 1.
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)0(curvemagnetic-N2.Figure 1
3.2. B-Magnetic Curves. In this section, we defined a new kind of magnetic curve called
B-magnetic curve in oriented 3D Manifolds, .,3 gM Moreover, we obtain somecharacterizations and examples of the curve.
Definition 3.2.Let 3: MI R be a curve in D3 oriented Riemannian space ),( 3 gM
and F be a magnetic field on .M We call the curve is a B-magnetic curve if the binormal
vector field of the curve satisfy the Lorentz force equation, that is,
(22) .)( BVBB
Proposition 3.7.Let be a unit speed B-magnetic curve in D3 oriented Riemannian space
),( 3 gM with the Frenet apparatus ,,,, BNT . Then we have the Serret-Frenet formulae:
(23) .
00
0
00
B
N
T
B
N
T
T
T
T
And then the Lorentz force in the Frenet frame written as
(24) .
00
0
00
)(
)(
)(
2
2
B
N
T
B
N
T
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where2
is a certain function.
Proof. Let be a unit speed N-magnetic curve in D3 oriented Riemannian space ),( 3 gM
with the Frenet apparatus ,,,, BNT . Since },,,{)( BNTspanT we have
BNTT )( and thus,
.0),(),()),(()),((
,)),((
,0)),((
2
TNgTBgTBgBTg
NTg
TTg
Therefore we can write
NT 2)(
Similarly, we can easily calculate that
.)(
)( 2
NB
BTN
These complete the proof.
Proposition 3.8.Let be a unit speed B-magnetic trajectory of a magnetic field V if and only
if V can be written along the curve as
(25) .2BTV
Proof. Let be a unit speed B-magnetic trajectory of a magnetic field .V Using the
Proposition 3.7 and Eq. (5), we can easily see that
.2BTV
This completes the proof.
Theorem 3.9.(main resul t) Let V be a Killing vector field on a simply connected space form
.),(3 gCM Then, the unit speed B-magnetic trajectories of VgCM ,),(3 are curves withcurvature and torsion satisfying
(26) .,0'''2)( 22 constCa
where C is curvature of the Riemannian space 3M and2
satisfy
(27) .2BTV
or
.)2
( BaTV
Proof. Let V be a magnetic field in a Riemannian 3D manifold. Then V satisfy Eq. (21).
Differentiating Eq. (21), we have
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(28) BNTVT
22)(
by differentiating of Eq. (29) with respect to s , we get
(29) BNTVT
)()2'()( 222
22
2
2
2
Proposition Pro 4 implies that ,0)( vV so from Eq. (a) and Eq. (29), we get
(30) 0
then, if Eq. (29) and Eq. (30) are considered with 0)( V in Proposition 2.2, we obtain
.0),),((2' 2 NTTVRg
In particular, if 3M has constant curvature ,C then 0),(),),(( NVCgNTTVRg and so
(31) .02'2
Similarly, if we combine Eq. (29) and Eq. (30) wih 0)(
V in Proposition 2.2, we have
.0),),((),),((222
2
2
2 BTTVRgBTTVRg
Hence, if 3M has constant curvature ,C then CBVCgBTTVRg ),(),),(( and
0),),(( BNTVRg gives
(32) .0222
2
2
2
C
Finally, considering Eq. (31)and Eq. (32) with Eq. (33), this implies
(33) .0'''2)( 22 Ca
Using Eq. (34), we obtain following second-order nonlinear ordinary differential equation
constants.and);()(,0)(2))(()()()(22 CstysCyasysytysy
Now, we consider the above differential equation in Euclidean 3 space ,E3 in 3 sphere3S
and in hyperbolic 3 space ,H3 respectively.
In D3 Euclidean space :E3
3,0),()(,0)1)((9)()()(2
Cssysysysysy
plots of sample indilidual solutions of this equation as:
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sample solution family:
space.Euclidean3Dincurvemagnetic-Bof)(curvaturetheofesTrajectori. t3Figure
In D3 sphere :S3
3,1),()(,0)(2)1)((9)()()( 2 Cssysysysysysy
plots of sample indilidual solutions of this equation as:
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sample solution family:
sphere.3Dincurvemagnetic-Bof)(curvaturetheofesTrajectori. t4Figure
In D3 hyperbolic space :H3
3,1),?()(,0)(2)1)((9)()()( 2 Cssysysysysysy
plots of sample indilidual solutions of this equation as:
sample solution family:
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space.Hyperbolic3Dincurvemagnetic-Bof)(curvaturetheofesTrajectori. t5Figure
Corollary 3.10. Let be a B-magnetic curve in a Euclidean 3-space with2
constant,
then is a general helix. Moreover, the axis of the helix is the vector field .V
Proof.
It is obvious from Eq. (23) and Eq. (21).
Corollary 3.11.The tangent indicatrix of the N-magnetic curve is a magnetic curve or B-magneticcurve in Euclidean 3-space.
Proof. In [14], we know that the tangent indicatrix of the slant helix is a general helix . So the proofis obvious from Corollary 3.5 and corollary 3.10.
Corollary 3.12.The binormal indicatrix of the N-magnetic curve is a circle.
Proof.In [14], we know that the binormal indicatrix of the slant helix is a circle . So the proof is
obvious from Corollary 3.5 and corollary 3.10.
Corollary 3.13.The normal indicatrix of the N-magnetic curve is a magnetic curve or B-magneticcurve in Euclidean 3-space.
Proof.In [14], we know that the normal indicatrix of the slant helix is a general helix . So the proofis obvious from Corollary 3.5 and corollary 3.10.
Example 3.3.We consider a N-magnetic curve in Euclidean 3 space is defined by
.15
)3cos(4),
64
2sin
64
9sin(
5
8),
64
2cos
64
9cos(
5
8)(
sssss
s
and the tangent indicatrix (see for details in [14]) of the curve is a general helix calculatedas
).3sin5
4,9cos
320
722cos
5
4),9sin
320
722sin
5
4(
)(
1)( sssss
s
s
The picture of the N-magnetic curve and its tangent indicatrix is rendered in Figure
6.
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ly.respective,indicatrixtangentitsandcurvemagnetic-N6.Figure
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