Maribor, July 1, 2008 2
Chaotic motion in rigid body Chaotic motion in rigid body dynamicsdynamics
7th International Summer School/Conference 7th International Summer School/Conference Let‘s face Chaos through Nonlinear DynamicsLet‘s face Chaos through Nonlinear Dynamics
CAMTP, University of MariborCAMTP, University of MariborJuly 1, 2008July 1, 2008
Peter H. RichterPeter H. Richter University of Bremen University of Bremen
Demo 2 - 4Demo 2 - 4
Maribor, July 1, 2008 3
OutlineOutline
• Parameter space• Configuration spaces SO(3) vs. T3
• Variations on Euler tops- with and without frame- effective potentials- integrable and chaotic dynamics
• Lagrange tops• Katok‘s family• Strategy of investigation
Thanks to my students Nils Keller and Konstantin FinkeThanks to my students Nils Keller and Konstantin Finke
Maribor, July 1, 2008 4
Parameter spaceParameter space
two moments of inertia
two angles for the center of gravity
at least one independent moment of inertia for the Cardan frame
angle between the frame‘s axis and the direction of gravity
6 essential parameters after scaling of lengths, time, energy:
Maribor, July 1, 2008 5
Configuration spaces SO(3) versus Configuration spaces SO(3) versus TT33
after separation of angle : reduced configuration spaces
Poisson ()-sphere Poisson ()-torus
„polar points“ defined with respect to an arbitrary direction
„polar -circles“ defined with respect to the axes of the framecoordinate singularities removed, but Euler variables lost
Euler angles ( ) Cardan angles ( )
Maribor, July 1, 2008 6
Demo 9, 10
surprise, surprise!surprise, surprise!
Maribor, July 1, 2008 7
Euler‘s top: no gravity, but torques by the Euler‘s top: no gravity, but torques by the frameframe
llzz
hh
Euler-Poisson )-torus
2222
2
cossin)cossin(2
z
clVcentrifugal
potential
2 S3
S1 x S2
Euler-Poisson )-sphere
EEReeb graph
Maribor, July 1, 2008 8
Nonsymmetric and symmetric Euler tops with Nonsymmetric and symmetric Euler tops with frameframe
Demo 5 - 8
33
33
33
integrable only if integrable only if the 3-axis is the 3-axis is symmetry axissymmetry axis
VB Euler
Maribor, July 1, 2008 9
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
Maribor, July 1, 2008 10
A nonintegrable Lagrange top with frameA nonintegrable Lagrange top with frame
pp = 7 = 7
pp = 6 = 6pp = 4.5 = 4.5pp = 3 = 3pp = 0 = 0
pp = 7.1 = 7.1 pp = 8 = 8 pp = 50 = 50
1 = 3 = 2.5 2 = 4.5R = 2.1 (s1, s2, s3) = (0, -1, 0)
8 types of effective potentials, depending on plz
Maribor, July 1, 2008 11
The Katok family – and othersThe Katok family – and othersarbitrary 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?
Maribor, July 1, 2008 12
Strategy of investigationStrategy of investigation
• search for critical points of effective potential Veff(; lz) no explicit general method seems to exist – numerical work required
• generate bifurcation diagrams in (h,lz)-plane• construct Reeb graphs• determine topology of energy surface for each connected component• for details of the foliation of energy surfaces look at Poincaré SoS:
as section condition take extrema of sz
project the surface of section onto the Poisson torus• accumulate knowledge and develop intuition for how chaos and order
are distributed in phase space and in parameter space
Maribor, July 1, 2008 14
Maribor, July 1, 2008 15
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
Maribor, July 1, 2008 17
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:
Maribor, July 1, 2008 18
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
Maribor, July 1, 2008 19
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
Maribor, July 1, 2008 20
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 spacespaceenergy 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
Maribor, July 1, 2008 21
Euler-Poisson equationsEuler-Poisson equations
coordinatescoordinates
Casimir constantsCasimir constants
effective potentialeffective potential
energy integralenergy integral
Maribor, July 1, 2008 22
Invariant sets in phase spaceInvariant sets in phase space
Maribor, July 1, 2008 23
(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 Ull0: ldU
Maribor, July 1, 2008 24
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
Maribor, July 1, 2008 25
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
Maribor, July 1, 2008 26
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
Maribor, July 1, 2008 27
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
Maribor, July 1, 2008 28
Enveloping surfacesEnveloping surfaces
BB
Maribor, July 1, 2008 29
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
Maribor, July 1, 2008 30
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
Maribor, July 1, 2008 31
EulerEuler LagrangeLagrange KovalevskayaKovalevskaya
Energy surfaces in action Energy surfaces in action representationrepresentation
Maribor, July 1, 2008 32
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
Maribor, July 1, 2008 33
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
Maribor, July 1, 2008 34
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
Maribor, July 1, 2008 35
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
Maribor, July 1, 2008 36
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)
Maribor, July 1, 2008 37
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 s ii = 0 (not done yet). = 0 (not done yet).
Maribor, July 1, 2008 38
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
Maribor, July 1, 2008 39
Poincaré section SPoincaré section S11
Skip 3Skip 3
Maribor, July 1, 2008 40
Poincaré section SPoincaré section S1 1 – projections to S– projections to S22(())
SS--
(())SS++
(())
00 0000
Maribor, July 1, 2008 41
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.
Maribor, July 1, 2008 42
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)
Maribor, July 1, 2008 43
Poincaré section SPoincaré section S22
=
Skip 3Skip 3
Maribor, July 1, 2008 44
Explicit formulae for the two sectionsExplicit formulae for the two sections
S1:with
S2:
where
Maribor, July 1, 2008 45
Poincaré sections SPoincaré sections S1 1 and Sand S22 in comparison in comparison
s =( 0.94868,0,0.61623)A =( 2, 1.1, 1)
Maribor, July 1, 2008 46
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
Maribor, July 1, 2008 47
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
Maribor, July 1, 2008 48
Example of a bifurcation scheme of periodic orbitsExample of a bifurcation scheme of periodic orbits
Maribor, July 1, 2008 49
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
Maribor, July 1, 2008 50
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