International Conference Classical Problems of Rigid Body Dynamics Donetsk, June 23-25, 2004 Peter H. Richter - Institut für Theoretische Physik The study of non-integrable rigid body problems S3S3S3S3 RP 3 K3K3K3K3 3S 3 (1.912,1.763)VII S 3,S 1 xS 2 2T 2

Donetsk, June 25, Types of rigid bodies: parameter space Translation R 2 or R 3 and Rotation SO(3) or T 3 4 parameters: principal moments of inertia: principal moments of inertia: one point fixed: only 3 rotational degrees of freedom center of gravity: in addition: moments of inertia and direction of axis of Cardan suspension 2 2

Donetsk, June 25, Rigid body dynamics - -Phase spaces and basic equations Full and reduced phase spaces Euler-Poisson equations - -Integrable cases Euler Lagrange Kovalevskaya - -Katok‘s general cases Effective potentials Bifurcation diagrams Enveloping surfaces - -Poincaré surfaces of section Gashenenko‘s version Dullin‘s suggestion Schmidt‘s investigations

Donetsk, June 25, Phase space and conserved quantities 3 angles + 3 velocities 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

Donetsk, June 25, Reduced phase space 3 components each of  and l 6D phase space but:     (Poisson sphere) und l ·  l (ang.momentum) are Casimirs  effectively only 4D phase space energy conservation h=const 3D energy surfaces integrable 2 conserved quantities 2D invariant sets super integrable 3D phs. sp., 2 integrals 1D invariant sets

Donetsk, June 25, Euler-Poisson equations coordinates Casimir constants effective potential energy integral

Donetsk, June 25, The basic scheme

Donetsk, June 25, Euler‘s case l- motion decouples from  -motion Poisson sphere potential admissible values in (p,q,r)-space (h,l)-bifurcation diagram

Donetsk, June 25, 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

Donetsk, June 25, Enveloping surfaces

Donetsk, June 25, 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

Donetsk, June 25, Effective potentials for case 1 (A1,A2,A3) = (1.7,0.9,0.86) l = 0l = 1.68l = 1.71l = 1.763l = l = 1.86l = 2.0l = 1.74

Donetsk, June 25, types of envelopes (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

Donetsk, June 25, types of envelopes (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)

Donetsk, June 25, 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 seems to complete the set of all possible types of envelopes.

Donetsk, June 25, Poincaré section S 1

Donetsk, June 25, Poincaré section S 1 – projections to S 2 (  ) S-()S-()S-()S-() S+()S+()S+()S+()  0       0 0

Donetsk, June 25, Poincaré section S 1 – polar circles Place the polar circles at upper and lower rims of the projection planes.

Donetsk, June 25, Poincaré section S 1 – projection artifacts s =( ,0, ) A =( 2, 1.1, 1)

Donetsk, June 25, Poincaré section S 2 =

Donetsk, June 25, Comparison of the two sections S1:S1: with S2:S2: where

Donetsk, June 25, Poincaré sections S 1 and S 2 in comparison s =( ,0, ) A =( 2, 1.1, 1)

Donetsk, June 25, 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

Donetsk, June 25, Examples B E

Donetsk, June 25, Example of a bifurcation scheme

Donetsk, June 25, Acknowledgement Igor Gashenenko Sven Schmidt Holger Dullin

Donetsk, June 25, Integrable cases Lagrange: „ heavy“, symmetric Kovalevskaya: Euler: „gravity-free“

Donetsk, June 25, Kovalevskaya‘s case (p,q,r)-equations integrals Tori projected to (p,q,r)-space Tori in phase space and Poincaré surface of section

Donetsk, June 25, Poincaré section S 1 – projections to S 2 (  ) S-()S-()S-()S-() S+()S+()S+()S+()  0       0 0

Donetsk, June 25, Poincaré section S 1 – polar circles

Donetsk, June 25, Improved projection representation Place the polar circles at upper and lower rims of the projection planes.