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Peter H. Richter - Institut für Theoretische PhysikPeter H. Richter - Institut für Theoretische Physik
6th International Summer School / Conference 6th International Summer School / Conference
„„Let‘s Face Chaos through Nonlinear Dynamics“ Let‘s Face Chaos through Nonlinear Dynamics“
CAMTP University of Maribor July 5, 2005CAMTP University of Maribor July 5, 2005
(1.912,1.763)VII
S3,S1xS2 2T2
Rigid Body DynamicsRigid Body Dynamics
SS33
RPRP33KK33
3S3S33
dedicated to my teacher
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Rigid bodies: parameter spaceRigid bodies: parameter space
Rotation SO(3)Rotation SO(3) or Tor T3 3 with one point fixedwith one point fixed
principal moments of inertia: principal moments of inertia: 321 AAA 1312 /,/ AAAA
321 ,, ssscenter of gravity:center of gravity:
With Cardan suspension, additional 2 parameters:With Cardan suspension, additional 2 parameters:
1 for moments of inertia and 1 for direction of axis1 for moments of inertia and 1 for direction of axis
22
,2 angles 2 angles
4 parameters:4 parameters:
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Rigid body dynamics in SO(3) Rigid body dynamics in SO(3)
- Phase spaces and basic equations• Full and reduced phase spaces• Euler-Poisson equations• Invariant sets and their bifurcations
- Integrable cases• Euler• Lagrange• Kovalevskaya
- Katok‘s more general cases• Effective potentials• Bifurcation diagrams• Enveloping surfaces
- Poincaré surfaces of section• Gashenenko‘s version• Dullin-Schmidt version• An application
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Phase space and conserved quantitiesPhase space and conserved quantities
3 angles + 3 momenta3 angles + 3 momenta 6D phase space6D phase space
energy conservation h=constenergy conservation h=const 5D energy surfaces 5D energy surfaces
one angular momentum l=constone angular momentum l=const 4D invariant sets4D invariant sets
3 conserved quantities3 conserved quantities 3D invariant sets3D invariant sets
4 conserved quantities4 conserved quantities 2D invariant sets2D invariant sets super-integrablesuper-integrable
integrableintegrable
mild chaosmild chaos
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Reduced phase spaceReduced phase space
The 6 components of The 6 components of and and ll are restricted by are restricted by
(Poisson sphere) and (Poisson sphere) and l l ··ll (angular (angular momentum) momentum) effectively only effectively only 4D phase 4D phase spacespace
energy conservation h=constenergy conservation h=const 3D energy surfaces3D energy surfaces
integrableintegrable2 conserved quantities2 conserved quantities 2D invariant sets2D invariant sets
super integrablesuper integrable 3 conserved quantities 3 conserved quantities 1D invariant sets1D invariant sets
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Euler-Poisson equationsEuler-Poisson equations
coordinatescoordinates
Casimir constantsCasimir constants
effective potentialeffective potential
energy integralenergy integral
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Invariant sets in phase spaceInvariant sets in phase space
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(h,l) bifurcation diagrams(h,l) bifurcation diagrams
)(3 R
0,0
0:),( dF
)(2 S
lU
),(),(: lhF
MomentumMomentum map map
EquivalentEquivalent statements: statements:
(h,l) is critical value(h,l) is critical value
relative equilibriumrelative equilibrium
is critical point of Uis critical point of U ll0: ldU
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Rigid body dynamics in SO(3)Rigid body dynamics in SO(3)
- Phase spaces and basic equations• Full and reduced phase spaces• Euler-Poisson equations• Invariant sets and their bifurcations
- Integrable cases• Euler• Lagrange• Kovalevskaya
- Katok‘s more general cases• Effective potentials• Bifurcation diagrams• Enveloping surfaces
- Poincaré surfaces of section• Gashenenko‘s version• Dullin-Schmidt version• An application
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Integrable casesIntegrable cases
Lagrange: Lagrange: „„heavy“, symmetricheavy“, symmetric
21 AA )1,0,0( s
Kovalevskaya: Kovalevskaya:
321 2AAA )0,0,1(s
Euler:Euler: „gravity-free“„gravity-free“
)0,0,0(s EE
LL
KK
AA
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Euler‘s caseEuler‘s case
ll--motionmotion decouples from decouples from --motionmotion
Poisson sphere potentialPoisson sphere potential
admissible values in (p,q,r)-space for given l and h < Uadmissible values in (p,q,r)-space for given l and h < U ll (h,l)-bifurcation diagram(h,l)-bifurcation diagram
BB
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Lagrange‘s caseLagrange‘s case
effective potentialeffective potential (p,q,r)-equations(p,q,r)-equations
integralsintegrals
I: ½ < I: ½ < < < ¾¾
II: ¾ < II: ¾ < < 1 < 1
RPRP33
bifurcation diagramsbifurcation diagrams
SS33
2S2S33
SS11xSxS22
III: III: > 1 > 1
SS11xSxS22
SS33 RPRP33
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Enveloping surfacesEnveloping surfaces
BB
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Kovalevskaya‘s caseKovalevskaya‘s case
(p,q,r)-equations(p,q,r)-equations
integralsintegrals
Tori projected Tori projected to (p,q,r)-spaceto (p,q,r)-space
Tori in phase space and Tori in phase space and Poincaré surface of sectionPoincaré surface of section
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Fomenko representation of foliations (3 examples out of 10)Fomenko representation of foliations (3 examples out of 10)
„„atoms“ of the atoms“ of the Kovalevskaya systemKovalevskaya system
elliptic center A elliptic center A
pitchfork bifurcation Bpitchfork bifurcation B
period doubling A* period doubling A*
double saddle Cdouble saddle C2 2
Critical tori: additional bifurcationsCritical tori: additional bifurcations
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EulerEuler LagrangeLagrange KovalevskayaKovalevskaya
Energy surfaces in action Energy surfaces in action representationrepresentation
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Rigid body dynamics in SO(3)Rigid body dynamics in SO(3)
- Phase spaces and basic equations• Full and reduced phase spaces• Euler-Poisson equations• Invariant sets and their bifurcations
- Integrable cases• Euler• Lagrange• Kovalevskaya
- Katok‘s more general cases• Effective potentials• Bifurcation diagrams• Enveloping surfaces
- Poincaré surfaces of section• Gashenenko‘s version• Dullin-Schmidt version• An application
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Katok‘s casesKatok‘s cases ss22 = s = s33 = 0 = 01
23
45 6
7
2
3
4 5 6 7
7 colors for 7 types of 7 colors for 7 types of bifurcation diagramsbifurcation diagrams
7colors for 7colors for 7 types of 7 types of energy energy surfacessurfaces
SS11xSxS22
1 2S2S33
SS33
RPRP33KK33
3S3S33
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Effective potentials for case 1Effective potentials for case 1 (A(A11,A,A22,A,A33) = (1.7,0.9,0.86)) = (1.7,0.9,0.86)
l = 1.763 l = 1.773 l = 1.86 l = 2.0
l = 0 l = 1.68 l = 1.71 l = 1.74
SS33
RPRP33KK33
3S3S33
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7+1 types of envelopes7+1 types of envelopes (I)(I) (A(A11,A,A22,A,A33) = (1.7,0.9,0.86)) = (1.7,0.9,0.86)
(h,l) = (1,1)I
S3 T2
(1,0.6)I‘
S3 T2
(2.5,2.15)II
2S3 2T2
(2,1.8)III
S1xS2 M32
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7+1 types of envelopes (II)7+1 types of envelopes (II)
(1.9,1.759)VI
3S3 2S2, T2
(1.912,1.763)VII
S3,S1xS2 2T2
IV
RP3 T2
(1.5,0.6) (1.85,1.705)V
K3 M32
(A(A11,A,A22,A,A33) = (1.7,0.9,0.86)) = (1.7,0.9,0.86)
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2 variations of types II and III2 variations of types II and III
2S3 2S2
II‘ (3.6,2.8)
S1xS2 T2
(3.6,2.75)III‘
Only in cases II‘ and III‘ are the envelopes free of singularities.
Case II‘ occurs in Katok‘s regions 4, 6, 7, case III‘ only in region 7.
A = (0.8,1.1,0.9)A = (0.8,1.1,0.9) A = (0.8,1.1,1.0)A = (0.8,1.1,1.0)
This completes the list of all possible This completes the list of all possible types of envelopes in the Katok case. types of envelopes in the Katok case. There are more in the more general There are more in the more general cases where only scases where only s33=0 (Gashenenko) =0 (Gashenenko)
or none of the sor none of the sii = 0 (not done yet). = 0 (not done yet).
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Rigid body dynamics in SO(3)Rigid body dynamics in SO(3)
- Phase spaces and basic equations• Full and reduced phase spaces• Euler-Poisson equations• Invariant sets and their bifurcations
- Integrable cases• Euler• Lagrange• Kovalevskaya
- Katok‘s more general cases• Effective potentials• Bifurcation diagrams• Enveloping surfaces
- Poincaré surfaces of section• Gashenenko‘s version• Dullin-Schmidt version• An application
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Poincaré section SPoincaré section S11
Skip 3Skip 3
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Poincaré section SPoincaré section S1 1 – projections to S– projections to S22(())
SS--
(())
SS++
(())
00 0000
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Poincaré section SPoincaré section S1 1 – polar circles– polar circles
)1,5.1,2(A
)0,0,1(s
Place the polar circles at Place the polar circles at upper and lower rims of the upper and lower rims of the projection planes. projection planes.
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Poincaré section SPoincaré section S1 1 – projection artifacts– projection artifacts
)1,1.1,2(A
)61623.0,0,94868.0(s
s =( 0.94868,0,0.61623)
A =( 2, 1.1, 1)
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Poincaré section SPoincaré section S22
=
Skip 3Skip 3
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Explicit formulae for the two sectionsExplicit formulae for the two sections
S1:with
S2:
where
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Poincaré sections SPoincaré sections S1 1 and Sand S22 in comparison in comparison
s =( 0.94868,0,0.61623)
A =( 2, 1.1, 1)
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From Kovalevskaya to LagrangeFrom Kovalevskaya to Lagrange(A(A11,A,A22,A,A33) = (2,) = (2,,1),1)
(s(s11,s,s22,s,s33) = (1,0,0)) = (1,0,0)
= 2 Kovalevskaya= 2 Kovalevskaya = 1.1 almost Lagrange= 1.1 almost Lagrange
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Examples: From Kovalevskaya to LagrangeExamples: From Kovalevskaya to Lagrange
BB EE
(A(A11,A,A22,A,A33) = (2,) = (2,,1),1)
(s(s11,s,s22,s,s33) = (1,0,0)) = (1,0,0)
= 2= 2 = 2= 2
= 1.1= 1.1 = 1.1= 1.1
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Example of a bifurcation scheme of periodic orbitsExample of a bifurcation scheme of periodic orbits
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To do listTo do list
• explore the chaosexplore the chaos
• work out the quantum mechanicswork out the quantum mechanics
• take frames into account take frames into account
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Thanks toThanks to
Holger Dullin Holger Dullin
Andreas WittekAndreas Wittek
Mikhail Kharlamov Mikhail Kharlamov
Alexey Bolsinov Alexey Bolsinov
Alexander Veselov Alexander Veselov
Igor GashenenkoIgor Gashenenko
Sven SchmidtSven Schmidt
… … and Siegfried Großmannand Siegfried Großmann
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