Groningen, June 3, From Spinning Tops to Rigid Body Motion Department of Mathematics, University of Groningen, June 3, 2009 Peter H. Richter University of Bremen S3S3 2S 3 S 1 xS 2

Groningen, June 3, Outline Demonstration of some basic physics Parameter sets Configuration spaces SO(3) and S 2 vs. T 3 and T 2 Phase space structure Equations of motion Strategies of investigation

Groningen, June 3, Demonstration of some basic physics Parameter sets Configuration spaces SO(3) and S 2 vs. T 3 and T 2 Phase space structure Equations of motion Strategies of investigation

Groningen, June 3, Parameter 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

Groningen, June 3, Demonstration of some basic physics Parameter sets Configuration spaces SO(3) and S 2 vs. T 3 and T 2 Phase space structure Equations of motion Strategies of investigation

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

Groningen, June 3, Demonstration of some basic physics Parameter sets Configuration spaces SO(3) and S 2 vs. T 3 and T 2 Phase space structure Equations of motion Strategies of investigation

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

Groningen, June 3, Reduced phase spaces with parameter l z 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

Groningen, June 3, (   l ) - phase space 3  i  -components + 3 momenta l i 6D phase space 3 conserved quantities 1D invariant sets super-integrable 2 Casimir constants  ·  = 1 and  ·l = l z 4D simplectic space 2 conserved quantities 2D invariant sets integrable energy conservation h = const 3D energy surfaces chaos

Groningen, June 3, Demonstration of some basic physics Parameter sets Configuration spaces SO(3) and S 2 vs. T 3 and T 2 Phase space structure Equations of motion Strategies of investigation

Groningen, June 3, Without frame: Euler-Poisson equations in ( ,l)-space Casimir constants: Coordinates: Energy constant: Effective potential:  motion:

Groningen, June 3, With frame: Euler – Lagrange equations where Reduction to a Hamiltonian with parameter, Coriolisforce and centrifugal potential Demo

Groningen, June 3, Demonstration of some basic physics Parameter sets Configuration spaces SO(3) and S 2 vs. T 3 and T 2 Phase space structure Equations of motion Strategies of investigation

Groningen, June 3, 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 Katok Envelope Actions Tori Poincaré Periods

Groningen, June 3, Katok‘s cases s 2 = s 3 = colors for 7 types of bifurcation diagrams 6colors for 6 types of energy surfaces S 1 xS 2 1 2S 3 S3S3 RP 3 K3K3 3S

Groningen, June 3, 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

Groningen, June 3, 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)

Groningen, June 3, EulerLagrangeKovalevskaya Energy surfaces in action representation

Groningen, June 3,

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

Groningen, June 3, Example of a bifurcation scheme of periodic orbits

Groningen, June 3, 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

Groningen, June 3, 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?

Groningen, June 3, Invariant sets in phase space

Groningen, June 3, (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

Groningen, June 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

Groningen, June 3, Integrable cases Lagrange: „ heavy“, symmetric Kovalevskaya: Euler: „gravity-free“ EEEE LLLL KKKK AAAA

Groningen, June 3, 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

Groningen, June 3, 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

Groningen, June 3, Enveloping surfaces BBBB

Groningen, June 3, Kovalevskaya‘s case (p,q,r)-equations integrals Tori projected to (p,q,r)-space Tori in phase space and Poincaré surface of section

Groningen, June 3, 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

Groningen, June 3, EulerLagrangeKovalevskaya Energy surfaces in action representation

Groningen, June 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

Groningen, June 3, 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

Groningen, June 3, 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

Groningen, June 3, 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

Groningen, June 3, 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)

Groningen, June 3, 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).

Groningen, June 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

Groningen, June 3, Poincaré section S 1 Skip 3 Skip 3

Groningen, June 3, Poincar é section S 1 – projections to S 2 (  ) S-()S-()S-()S-() S+()S+()S+()S+()  0       0 0

Groningen, June 3, Poincaré section S 1 – polar circles Place the polar circles at upper and lower rims of the projection planes.

Groningen, June 3, Poincaré section S 1 – projection artifacts s =( ,0, ) A =( 2, 1.1, 1)

Groningen, June 3, Explicit formulae for the two sections S1:S1: with S2:S2: where

Groningen, June 3, Poincaré sections S 1 and S 2 in comparison s =( ,0, ) A =( 2, 1.1, 1)

Groningen, June 3, 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

Groningen, June 3, 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

Groningen, June 3, Example of a bifurcation scheme of periodic orbits