2. Maribor, July 1, 2008 2 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

3. Maribor, July 1, 2008 3 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

4. Maribor, July 1, 2008 4 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:

5. Maribor, July 1, 2008 5 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 (     ) 

6. Maribor, July 1, 2008 6 Demo 9, 10 surprise, surprise!

7. Maribor, July 1, 2008 7 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

8. Maribor, July 1, 2008 8 Nonsymmetric and symmetric Euler tops with frame Demo 5 - 8 3 3 3 integrable only if the 3-axis is symmetry axis VB Euler

9. Maribor, July 1, 2008 9 Lagrange tops without frame Three types of bifurcation diagrams: 0.5 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

10. Maribor, July 1, 2008 10 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

11. Maribor, July 1, 2008 11 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?

12. Maribor, July 1, 2008 12 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

14. Maribor, July 1, 2008 14

15. Maribor, July 1, 2008 15

16. 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

17. Maribor, July 1, 2008 17 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:

18. Maribor, July 1, 2008 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 1, 2008 19 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

20. Maribor, July 1, 2008 20 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

21. Maribor, July 1, 2008 21 Euler-Poisson equations coordinates Casimir constants effective potential energy integral

22. Maribor, July 1, 2008 22 Invariant sets in phase space

23. Maribor, July 1, 2008 23 (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

24. Maribor, July 1, 2008 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 1, 2008 25 Integrable cases Lagrange: „ heavy“, symmetric Kovalevskaya: Euler: „gravity-free“ EEEE LLLL KKKK AAAA

26. Maribor, July 1, 2008 26 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

27. Maribor, July 1, 2008 27 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

28. Maribor, July 1, 2008 28 Enveloping surfaces BBBB

29. Maribor, July 1, 2008 29 Kovalevskaya‘s case (p,q,r)-equations integrals Tori projected to (p,q,r)-space Tori in phase space and Poincaré surface of section

30. Maribor, July 1, 2008 30 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

31. Maribor, July 1, 2008 31 EulerLagrangeKovalevskaya Energy surfaces in action representation

32. Maribor, July 1, 2008 32 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

33. Maribor, July 1, 2008 33 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

34. Maribor, July 1, 2008 34 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

35. Maribor, July 1, 2008 35 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

36. Maribor, July 1, 2008 36 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)

37. Maribor, July 1, 2008 37 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).

38. Maribor, July 1, 2008 38 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

39. Maribor, July 1, 2008 39 Poincaré section S 1 Skip 3 Skip 3

40. Maribor, July 1, 2008 40 Poincar é section S 1 – projections to S 2 (  ) S-()S-()S-()S-() S+()S+()S+()S+()  0       0 0

41. Maribor, July 1, 2008 41 Poincaré section S 1 – polar circles Place the polar circles at upper and lower rims of the projection planes.

42. Maribor, July 1, 2008 42 Poincaré section S 1 – projection artifacts s =( 0.94868,0,0.61623) A =( 2, 1.1, 1)

43. Maribor, July 1, 2008 43 Poincaré section S 2 = Skip 3 Skip 3

44. Maribor, July 1, 2008 44 Explicit formulae for the two sections S1:S1: with S2:S2: where

45. Maribor, July 1, 2008 45 Poincaré sections S 1 and S 2 in comparison s =( 0.94868,0,0.61623) A =( 2, 1.1, 1)

46. Maribor, July 1, 2008 46 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

47. Maribor, July 1, 2008 47 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

48. Maribor, July 1, 2008 48 Example of a bifurcation scheme of periodic orbits

49. Maribor, July 1, 2008 49 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

50. Maribor, July 1, 2008 50 Thanks to Holger Dullin Andreas Wittek Mikhail Kharlamov Alexey Bolsinov Alexander Veselov Igor Gashenenko Sven Schmidt … and Siegfried Großmann