Maribor, July 1, Chaotic motion in rigid body dynamics 7th International Summer School/Conference Let‘s face Chaos through Nonlinear Dynamics CAMTP, University of Maribor July 1, 2008 Peter H. Richter University of Bremen Demo 2 - 4

Maribor, July 1, Outline Parameter space Configuration spaces SO(3) vs. T 3 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 Finke

Maribor, July 1, Parameter 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, Configuration spaces SO(3) versus T 3 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 frame coordinate singularities removed, but Euler variables lost Euler angles (     ) Cardan angles (     ) 

Maribor, July 1, Demo 9, 10 surprise, surprise!

Maribor, July 1, Euler‘s top: no gravity, but torques by the frame lzlzlzlzh Euler-Poisson   )-torus          centrifugal potential 2 S 3 S 1 x S 2 Euler-Poisson   )-sphere E Reeb graph

Maribor, July 1, Nonsymmetric and symmetric Euler tops with frame Demo integrable only if the 3-axis is symmetry axis VB Euler

Maribor, July 1, Lagrange tops without frame Three types of bifurcation diagrams: (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, A nonintegrable Lagrange top with frame p  = 7 p  = 6 p  = 4.5 p  = 3 p  = 0 p  = 7.1 p  = 8 p  = 50  1 =  3 = 2.5  2 = 4.5  R = 2.1 (s 1, s 2, s 3 ) = (0, -1, 0) 8 types of effective potentials, depending on p    l z

Maribor, July 1, The 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?

Maribor, July 1, Strategy of investigation search for critical points of effective potential V eff (  ; l z ) no explicit general method seems to exist – numerical work required generate bifurcation diagrams in (h,l z )-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 s z 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,

Maribor, July 1,

6th International Summer School / Conference „Let‘s Face Chaos through Nonlinear Dynamics“ CAMTP University of Maribor July 5, 2005 Peter H. Richter - Institut für Theoretische Physik (1.912,1.763)VII S 3,S 1 xS 2 2T 2 Rigid Body Dynamics S3S3S3S3 RP 3 K3K3K3K3 3S 3 dedicated to my teacher

Maribor, July 1, Rigid bodies: parameter space Rotation SO(3) or T 3 with one point fixed principal moments of inertia: center of gravity: With Cardan suspension, additional 2 parameters: 1 for moments of inertia and 1 for direction of axis 2 2 angles 4 parameters:

Maribor, July 1, 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, Phase space and conserved quantities 3 angles + 3 momenta 6D phase space energy conservation h=const 5D energy surfaces one angular momentum l=const 4D invariant sets 3 conserved quantities 3D invariant sets 4 conserved quantities 2D invariant sets super-integrable integrable mild chaos

Maribor, July 1, Reduced phase space The 6 components of  and l are restricted by     (Poisson sphere) and l ·  l (angular momentum)  effectively only 4D phase space energy conservation h=const 3D energy surfaces integrable 2 conserved quantities 2D invariant sets super integrable 3 conserved quantities 1D invariant sets

Maribor, July 1, Euler-Poisson equations coordinates Casimir constants effective potential energy integral

Maribor, July 1, Invariant sets in phase space

Maribor, July 1, (h,l) bifurcation diagrams Momentum map Equivalent statements: (h,l) is critical value relative equilibrium is critical point of U l  is critical point of U l

Maribor, July 1, 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, Integrable cases Lagrange: „ heavy“, symmetric Kovalevskaya: Euler: „gravity-free“ EEEE LLLL KKKK AAAA

Maribor, July 1, Euler‘s case l- motion decouples from  -motion Poisson sphere potential admissible values in (p,q,r)-space for given l and h < U l (h,l)-bifurcation diagram BBBB

Maribor, July 1, Lagrange‘s case effective potential (p,q,r)-equations integrals I: ½ <  < ¾ II: ¾ <  < 1 RP 3 bifurcation diagrams S3S3S3S3 2S 3 S 1 xS 2 III:  > 1 S 1 xS 2 S3S3S3S3 RP 3

Maribor, July 1, Enveloping surfaces BBBB

Maribor, July 1, Kovalevskaya‘s case (p,q,r)-equations integrals Tori projected to (p,q,r)-space Tori in phase space and Poincaré surface of section

Maribor, July 1, Fomenko representation of foliations (3 examples out of 10) „atoms“ of the Kovalevskaya system elliptic center A pitchfork bifurcation B period doubling A* double saddle C 2 Critical tori: additional bifurcations

Maribor, July 1, EulerLagrangeKovalevskaya Energy surfaces in action representation

Maribor, July 1, 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, Katok‘s cases s 2 = s 3 = colors for 7 types of bifurcation diagrams 7colors for 7 types of energy surfaces S 1 xS 2 1 2S 3 S3S3S3S3 RP 3 K3K3K3K3 3S 3

Maribor, July 1, Effective potentials for case 1 (A 1,A 2,A 3 ) = (1.7,0.9,0.86) l = 1.763l = l = 1.86l = 2.0 l = 0l = 1.68l = 1.71 l = 1.74 S3S3S3S3 RP 3 K3K3K3K3 3S 3

Maribor, July 1, types of envelopes (I) (A 1,A 2,A 3 ) = (1.7,0.9,0.86) (h,l) = (1,1) I S3S3 T2T2 (1,0.6) I‘ S3S3 T2T2 (2.5,2.15) II 2S 3 2T 2 (2,1.8) III S 1 xS 2 M32M32

Maribor, July 1, types of envelopes (II) (1.9,1.759) VI 3S 3 2S 2, T 2 (1.912,1.763)VII S 3,S 1 xS 2 2T 2 IV RP 3 T2T2 (1.5,0.6) (1.85,1.705) V K3K3 M32M32 (A 1,A 2,A 3 ) = (1.7,0.9,0.86)

Maribor, July 1, variations of types II and III 2S 3 2S 2 II‘ (3.6,2.8) S 1 xS 2 T2T2 (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 = (0.8,1.1,1.0) This completes the list of all possible types of envelopes in the Katok case. There are more in the more general cases where only s 3 =0 (Gashenenko) or none of the s i = 0 (not done yet).

Maribor, July 1, 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, Poincaré section S 1 Skip 3 Skip 3

Maribor, July 1, Poincar é section S 1 – projections to S 2 (  ) S-()S-()S-()S-() S+()S+()S+()S+()  0       0 0

Maribor, July 1, Poincaré section S 1 – polar circles Place the polar circles at upper and lower rims of the projection planes.

Maribor, July 1, Poincaré section S 1 – projection artifacts s =( ,0, ) A =( 2, 1.1, 1)

Maribor, July 1, Poincaré section S 2 = Skip 3 Skip 3

Maribor, July 1, Explicit formulae for the two sections S1:S1: with S2:S2: where

Maribor, July 1, Poincaré sections S 1 and S 2 in comparison s =( ,0, ) A =( 2, 1.1, 1)

Maribor, July 1, From Kovalevskaya to Lagrange (A 1,A 2,A 3 ) = (2, ,1) (s 1,s 2,s 3 ) = (1,0,0) (s 1,s 2,s 3 ) = (1,0,0)  = 2 Kovalevskaya  = 1.1 almost Lagrange

Maribor, July 1, Examples: From Kovalevskaya to Lagrange B E (A 1,A 2,A 3 ) = (2, ,1) (s 1,s 2,s 3 ) = (1,0,0) (s 1,s 2,s 3 ) = (1,0,0)  = 2  = 1.1

Maribor, July 1, Example of a bifurcation scheme of periodic orbits

Maribor, July 1, To do list explore the chaos explore the chaos work out the quantum mechanics work out the quantum mechanics take frames into account take frames into account

Maribor, July 1, Thanks to Holger Dullin Andreas Wittek Mikhail Kharlamov Alexey Bolsinov Alexander Veselov Igor Gashenenko Sven Schmidt … and Siegfried Großmann