2. 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

3. Maribor, July 5, 2005 3 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:

4. Maribor, July 5, 2005 4 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

5. Maribor, July 5, 2005 5 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

6. Maribor, July 5, 2005 6 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

7. Maribor, July 5, 2005 7 Euler-Poisson equations coordinates Casimir constants effective potential energy integral

8. Maribor, July 5, 2005 8 Invariant sets in phase space

9. Maribor, July 5, 2005 9 (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

10. Maribor, July 5, 2005 10 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

11. Maribor, July 5, 2005 11 Integrable cases Lagrange: „ heavy“, symmetric Kovalevskaya: Euler: „gravity-free“ EEEE LLLL KKKK AAAA

12. Maribor, July 5, 2005 12 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

13. Maribor, July 5, 2005 13 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

14. Maribor, July 5, 2005 14 Enveloping surfaces BBBB

15. Maribor, July 5, 2005 15 Kovalevskaya‘s case (p,q,r)-equations integrals Tori projected to (p,q,r)-space Tori in phase space and Poincaré surface of section

16. Maribor, July 5, 2005 16 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

17. Maribor, July 5, 2005 17 EulerLagrangeKovalevskaya Energy surfaces in action representation

18. Maribor, July 5, 2005 18 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

19. Maribor, July 5, 2005 19 Katok‘s cases s 2 = s 3 = 0 1 2 3 4 56 7 2 3 45 6 7 7 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

20. Maribor, July 5, 2005 20 Effective potentials for case 1 (A 1,A 2,A 3 ) = (1.7,0.9,0.86) l = 1.763l = 1.773 l = 1.86l = 2.0 l = 0l = 1.68l = 1.71 l = 1.74 S3S3S3S3 RP 3 K3K3K3K3 3S 3

21. Maribor, July 5, 2005 21 7+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

22. Maribor, July 5, 2005 22 7+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)

23. Maribor, July 5, 2005 23 2 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).

24. Maribor, July 5, 2005 24 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

25. Maribor, July 5, 2005 25 Poincaré section S 1 Skip 3 Skip 3

26. Maribor, July 5, 2005 26 Poincaré section S 1 – projections to S 2 (  ) S-()S-()S-()S-() S+()S+()S+()S+()  0       0 0

27. Maribor, July 5, 2005 27 Poincaré section S 1 – polar circles Place the polar circles at upper and lower rims of the projection planes.

28. Maribor, July 5, 2005 28 Poincaré section S 1 – projection artifacts s =( 0.94868,0,0.61623) A =( 2, 1.1, 1)

29. Maribor, July 5, 2005 29 Poincaré section S 2 = Skip 3 Skip 3

30. Maribor, July 5, 2005 30 Explicit formulae for the two sections S1:S1: with S2:S2: where

31. Maribor, July 5, 2005 31 Poincaré sections S 1 and S 2 in comparison s =( 0.94868,0,0.61623) A =( 2, 1.1, 1)

32. Maribor, July 5, 2005 32 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

33. Maribor, July 5, 2005 33 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

34. Maribor, July 5, 2005 34 Example of a bifurcation scheme of periodic orbits

35. Maribor, July 5, 2005 35 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

36. Maribor, July 5, 2005 36 Thanks to Holger Dullin Andreas Wittek Mikhail Kharlamov Alexey Bolsinov Alexander Veselov Igor Gashenenko Sven Schmidt … and Siegfried Großmann