2. Groningen, June 3, 2009 2 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

3. Groningen, June 3, 2009 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

4. Groningen, June 3, 2009 4 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

5. Groningen, June 3, 2009 5 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

6. Groningen, June 3, 2009 6 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

7. Groningen, June 3, 2009 7 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 (     )

8. Groningen, June 3, 2009 8 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

9. Groningen, June 3, 2009 9 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

10. Groningen, June 3, 2009 10 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

11. Groningen, June 3, 2009 11 (   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

12. Groningen, June 3, 2009 12 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

13. Groningen, June 3, 2009 13 Without frame: Euler-Poisson equations in ( ,l)-space Casimir constants: Coordinates: Energy constant: Effective potential:  motion:

14. Groningen, June 3, 2009 14 With frame: Euler – Lagrange equations where Reduction to a Hamiltonian with parameter, Coriolisforce and centrifugal potential Demo

15. Groningen, June 3, 2009 15 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

16. Groningen, June 3, 2009 16 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

18. Groningen, June 3, 2009 18 Katok‘s cases s 2 = s 3 = 0 1 3 5 2 3 45 6 7 7 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 3 2 4 6 7 1 3 5

19. Groningen, June 3, 2009 19 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

20. Groningen, June 3, 2009 20 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)

21. Groningen, June 3, 2009 21 EulerLagrangeKovalevskaya Energy surfaces in action representation

22. Groningen, June 3, 2009 22

23. Groningen, June 3, 2009 23 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

24. Groningen, June 3, 2009 24 Example of a bifurcation scheme of periodic orbits

26. Groningen, June 3, 2009 26 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

27. Groningen, June 3, 2009 27 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?

28. Groningen, June 3, 2009 28 Invariant sets in phase space

29. Groningen, June 3, 2009 29 (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

30. Groningen, June 3, 2009 30 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

31. Groningen, June 3, 2009 31 Integrable cases Lagrange: „ heavy“, symmetric Kovalevskaya: Euler: „gravity-free“ EEEE LLLL KKKK AAAA

32. Groningen, June 3, 2009 32 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

33. Groningen, June 3, 2009 33 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

34. Groningen, June 3, 2009 34 Enveloping surfaces BBBB

35. Groningen, June 3, 2009 35 Kovalevskaya‘s case (p,q,r)-equations integrals Tori projected to (p,q,r)-space Tori in phase space and Poincaré surface of section

36. Groningen, June 3, 2009 36 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

37. Groningen, June 3, 2009 37 EulerLagrangeKovalevskaya Energy surfaces in action representation

38. Groningen, June 3, 2009 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. Groningen, June 3, 2009 39 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

40. Groningen, June 3, 2009 40 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

41. Groningen, June 3, 2009 41 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

42. Groningen, June 3, 2009 42 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)

43. Groningen, June 3, 2009 43 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).

44. Groningen, June 3, 2009 44 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

45. Groningen, June 3, 2009 45 Poincaré section S 1 Skip 3 Skip 3

46. Groningen, June 3, 2009 46 Poincar é section S 1 – projections to S 2 (  ) S-()S-()S-()S-() S+()S+()S+()S+()  0       0 0

47. Groningen, June 3, 2009 47 Poincaré section S 1 – polar circles Place the polar circles at upper and lower rims of the projection planes.

48. Groningen, June 3, 2009 48 Poincaré section S 1 – projection artifacts s =( 0.94868,0,0.61623) A =( 2, 1.1, 1)

49. Groningen, June 3, 2009 49 Explicit formulae for the two sections S1:S1: with S2:S2: where

50. Groningen, June 3, 2009 50 Poincaré sections S 1 and S 2 in comparison s =( 0.94868,0,0.61623) A =( 2, 1.1, 1)

51. Groningen, June 3, 2009 51 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

52. Groningen, June 3, 2009 52 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

53. Groningen, June 3, 2009 53 Example of a bifurcation scheme of periodic orbits