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Groningen, June 3, 2009 2
From Spinning Tops to Rigid Body Motion
Department of Mathematics, University of Groningen, June 3, 2009
Peter H. Richter University of Bremen S3
2S3
S1xS2
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OutlineOutline
• Demonstration of some basic physics• Parameter sets • Configuration spaces SO(3) and S2 vs. T3 and T2
• Phase space structure• Equations of motion• Strategies of investigation
Groningen, June 3, 2009 4
• Demonstration of some basic physics• Parameter sets • Configuration spaces SO(3) and S2 vs. T3 and T2
• Phase space structure• Equations of motion• Strategies of investigation
Groningen, June 3, 2009 5
Parameter spaceParameter space
at least one independent moment of inertia for the Cardan frame
6 essential parameters after scaling of lengths, time, energy:
angle between the frame‘s axis and the direction of gravity
two moments of inertia
two angles for the center of gravity
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• Demonstration of some basic physics• Parameter sets • Configuration spaces SO(3) and S2 vs. T3 and T2
• Phase space structure• Equations of motion• Strategies of investigation
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Configuration spaces SO(3) versus Configuration spaces SO(3) versus TT33
after separation of angle : reduced configuration spaces
Poisson ()-sphere
„polar points“ defined with respect to an arbitrary direction
Poisson ()-torus
„polar -circles“ defined with respect to the axes of the framecoordinate singularities removed, but Euler variables lost
Cardan angles ( )
Euler angles ( )
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• Demonstration of some basic physics• Parameter sets • Configuration spaces SO(3) and S2 vs. T3 and T2
• Phase space structure • Equations of motion • Strategies of investigation
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Phase space and conserved quantities
3 angles + 3 momenta 6D phase space
4 conserved quantities 2D invariant sets super-integrable
one angular momentum lz = const 4D invariant sets mild chaos
energy conservation h = const 5D energy surfaces strong chaos
3 conserved quantities 3D invariant sets integrable
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Reduced phase spaces with parameter lz
2 angles + 2 momenta 4D phase space
3 conserved quantities 1D invariant sets super-integrable
2 conserved quantities 2D invariant sets integrable
energy conservation h = const 3D energy surfaces chaos
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( l ) - phase space
3 i-components + 3 momenta li 6D phase space
3 conserved quantities 1D invariant sets super-integrable
2 Casimir constants · = 1 and ·l = lz 4D simplectic space
2 conserved quantities 2D invariant sets integrable
energy conservation h = const 3D energy surfaces chaos
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• Demonstration of some basic physics• Parameter sets • Configuration spaces SO(3) and S2 vs. T3 and T2
• Phase space structure • Equations of motion• Strategies of investigation
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Without frame: Euler-Poisson equations in (,l)-space
1 zll Casimir constants: Casimir constants:
),,( 321
),,( 321
AAAAllll ),,(),,( 332211321
Coordinates: Coordinates:
Energy constant: Energy constant: slA
lA
lA
h 23
3
22
2
21
1 2
1
2
1
2
1
Effective potential: Effective potential: sAAA
lU zl
)(2 233
222
211
2
motion: motion: 22
21
2211
sll
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With frame: Euler – Lagrange equations
2)cos sin sin (2
1 T
2)sin cos sin (2
1
22
2
1)cos (
2
1
)cos ,sin sin ,cos sin( s
)0( cos cos sin )(sin sin V
wherewhere
Reduction to a Hamiltonian with parameter , Coriolisforce and centrifugal potential
),,,( ppHH zlp
zl 2zl
DemoDemo
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• Demonstration of some basic physics• Parameter sets • Configuration spaces SO(3) and S2 vs. T3 and T2
• Phase space structure • Equations of motion • Strategies of investigation
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• topological bifurcations of iso-energy surfaces
• their projections to configuration and momentum spaces• integrable systems: action variable representation and
foliation by invariant tori
• chaotic systems: Poincaré sections • periodic orbit skeleton: stable (order) and unstable (chaos)
Search for invariant sets in phase space, and their bifurcations
KatokKatok
EnvelopeEnvelope
ActionsActions
ToriTori
PoincaréPoincaré
PeriodsPeriods
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Katok‘s cases s2 = s3 = 01
3
5
2
3
4 5 6 7
7 colors for 7 types of bifurcation diagrams
6colors for 6 types of energy surfaces
S1xS2
1 2S3
S3
RP3K3
3S3
24
6
7
13
5
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7+1 types of envelopes (I) (A1,A2,A3) = (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)
(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
(A1,A2,A3) = (1.7,0.9,0.86)
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Euler Lagrange Kovalevskaya
Energy surfaces in action representation
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Examples: From Kovalevskaya to Lagrange
BB EE
(A1,A2,A3) = (2,,1)
(s1,s2,s3) = (1,0,0)
= 2 = 2
= 1.1 = 1.1
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Example of a bifurcation scheme of periodic orbits
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Lagrange tops without frameLagrange tops without frame
Three types of bifurcation diagrams:
0.5 < < 0.75 (discs), 0.75 < < 1 (balls), > 1 (cigars)
five types of Reeb graphs
When the 3-axis is the symmetry axis, the system remains integrable with the frame, otherwise not.
VB Lagrange
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The Katok family – and othersThe Katok family – and others
arbitrary moments of inertia, (s1, s2, s3) = (1, 0, 0)
Topology of 3D energy surfaces and 2D Poincaré surfaces of section has been analyzed completely (I. N. Gashenenko, P. H. R. 2004)
How is this modified by the Cardan frame?
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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
Groningen, June 3, 2009 42
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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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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