1. Peter Richter Institute for Theoretical Physics 1 Integrable and Non-Integrable Rigid Body Dynamics – Actions and Poincare Sections Classical Problems of Rigid Body Dynamics International Conference Dedicated to the 300th anniversary of Leonhard Euler Donetsk 9-13 June 2007

2. Peter Richter Institute for Theoretical Physics 2 Parameter space of standard rigid bodies No translation 3 rotational degrees of freedom of SO(3) type Remark: a Cardan suspension implies T 3 rather than SO(3) as configurations space; it requires at least two more parameters for the moment of inertia of the suspension device, and the direction of its axis relative to gravity. 4 parameters principal moments of inertia A 1, A 2, A 3 2:  = A 2 /A 1,  = A 3 /A 1 center of gravity s 1, s 2, s 3 2: 

3. Peter Richter Institute for Theoretical Physics 3 Phase spaces and basic equations -Full and reduced phase spaces -Equations of motion Integrable cases: Actions -Euler -Lagrange -Kovalevskaya Non-integrable dynamics: Poincaré sections -General principles -The PP-torus representation -An example Integrable and non-integrable rigid body dynamics

4. Peter Richter Institute for Theoretical Physics 4 Phase space and conserved quantities 3 angles + 3 velocities 6D phase space 3 conserved quantities: energy h, momenta l z, l 3 → 3D invariant sets, integrable dynamics 4 conserved quantities: energy h, angular momenta l x, l y, l z → 2D invariant sets, super-integrable dynamics 3 conserved quantities: energy h, momentum l z, and K → 3D invariant sets, integrable In general: only 2 conserved quantities: energy h and momentum l z → 4D invariant sets, (mildly) chaotic dynamics

5. Peter Richter Institute for Theoretical Physics 5 Reduced phase space 2 conserved quantities: energy h, momentum l 3 → 2D invariant sets, integrable dynamics 3 conserved quantities: energy h, angular momenta l x, l y → 1D invariant sets, super-integrable dynamics 2 conserved quantities: energy h, Kovalevskaya K → 2D invariant sets, integrable In general: only 1 conserved quantity: energy h → 3D invariant sets, (mildly) chaotic dynamics The 6D phase space of variables  =(  1,  2,  3 ) and l = (l 1,l 2,l 3 ) has Casimir constants  2 =1 (Poisson sphere) and l·  = l z, hence is effectively 4D

6. Peter Richter Institute for Theoretical Physics 6 Euler-Poisson equations coordinates general constant: energy phase space with Poisson structure and 2 Casimirs Hamiltonian equations of motion effective potential

7. Peter Richter Institute for Theoretical Physics 7 Canonical equations with Euler angles phase space coordinates Ha miltonians: Euler‘s case Lagrange‘s case

8. Peter Richter Institute for Theoretical Physics 8 Phase spaces and basic equations -Full and reduced phase spaces -Equations of motion Integrable cases: Actions -Euler -Lagrange -Kovalevskaya Non-integrable dynamics: Poincaré sections -General principles -The PP-torus representation -An example Integrable and non-integrable rigid body dynamics

9. Peter Richter Institute for Theoretical Physics 9 Actions in the Euler case actions  -action:  -actions: complete elliptic integrals of the third kind Energy surfaces h = h(I 1,I 2,I 3 ) carry information on frequencies, winding numbers, bifurcations (separatrices), and quantum mechanics.

10. Peter Richter Institute for Theoretical Physics 10 Actions in the Euler case  1 =  2  =  3  1 = 1.5,  2  = 2  3 = 1 I  towards left front, I  towards right, I pointing up only one quarter is shown: I  and I  may be positive and negative, depending on sense of rotation Energy surfaces in action representation: 1:1 correspondence of points and Liouville tori

11. Peter Richter Institute for Theoretical Physics 11 Actions in the Lagrange case actions  -action: Frequencies are integrals of the first kind, winding numbers are of the third kind  -action: complete elliptic integral of the third kind -action:

12. Peter Richter Institute for Theoretical Physics 12 Actions in the Lagrange case I  towards right, I  towards back, I pointing up only one half is shown: surfaces are symmetric with respect to I , I  → -I , -I     =    =      =    =      =    =      =   E = 0 E = 1 E = 1.5 E = 4

13. Peter Richter Institute for Theoretical Physics 13 Actions in the Kovalevskaya case s 1,s 2 -actions:  -action: I 1 = l  = l hyperelliptic integral with paths to be determined on a hyperelliptic curve s-coordinate w.r.t. pole h w.r.t. Appelrot-transition Kovalevskaya‘s polynomials: dd

14. Peter Richter Institute for Theoretical Physics 14 Actions in the Kovalevskaya case Energy surfaces in action representation: agreement of numerical computation with Abelian integrals I  towards right front, I 2 towards left front, I 3 pointing up h = -0.5 h = 0.9 h = 1.2h = 1.45 h = 2.3 h = 9.0

15. Peter Richter Institute for Theoretical Physics 15 Phase spaces and basic equations -Full and reduced phase spaces -Equations of motion Integrable cases: Actions -Euler -Lagrange -Kovalevskaya Non-integrable dynamics: Poincaré sections -General principles -The PP-torus representation -An example Integrable and non-integrable rigid body dynamics

16. Peter Richter Institute for Theoretical Physics 16 Poincaré surfaces of section: general considerations Find a surface P 2 h,l in E 3 h,l  such that every trajectory meets it repeatedly. Recipe: choose bounded W: E 3 h,l → R and take S := dW/dt = 0 as section condition. Then look for a convenient surface to which P 2 h,l can be mapped 1:1. P 2 h,l S+()S+() S─()S─() Finally, identify points on S + (  ) and S ─ (  ) whose preimages are identical.

17. Peter Richter Institute for Theoretical Physics 17 Implementation of the general principle First idea: divide P 2 h,l according to incoming and outgoing intersections dS/dt 0, respectively. Problem: Projections in general not 1:1 dS/dt = 0 det  H,L z,S) /  l = 0 Second idea: divide at points where the projection has less than full rank. Then each half may have incoming and outgoing intersections, but so what?

18. Peter Richter Institute for Theoretical Physics 18 The situation in rigid body dynamics Let the section condition be defined by W =  ∙ s, hence S = A -1 l ∙ (s x  ) = 0. The surface P 2 h,l so defined projects to the Poisson sphere as follows: Its image is the entire accessible region U h,l = { U l (  ) ≤ h } in S 2 (  ). Points on the boundary  U h,l have one preimage. The preimage of the two points  = ± s /s is a circle. All other points of U h,l have exactly two preimages. This suggests the construction of the PP-torus (from Poisson and Poincaré).

19. Peter Richter Institute for Theoretical Physics 19 The PP-torus S ─ (  ) C ─ (  ) S + (  ) C + (  ) T 2 PP (  ) A = (2,1.5,1) s = (1,0,0) h = 80.5 l = 12.8 P 2 h,l ~ M 2 3 in out in out n.acc.

20. Peter Richter Institute for Theoretical Physics 20 An example A =(2,1.1,1) s = (0.94868, 0, 0.61623) l = 3.25 h = 1.8 h = 2.6 S 2S 2 T 2

21. Peter Richter Institute for Theoretical Physics 21 h = 3.8 h = 4.45 2 T 2 T 2 + S 2

22. Peter Richter Institute for Theoretical Physics 22 h = 5.3 h = 4.675 M 2 3 M 2 2

23. Peter Richter Institute for Theoretical Physics 23 h = 6.5, l = 3.25 h = 14.0, l = 7.7 h = 3.1486, l = 2.72 M 2 2 2 S 2 T 2

24. Peter Richter Institute for Theoretical Physics 24 -Phase spaces and basic equations Full and reduced phase spaces Equations of motion -Integrable cases: Actions Euler Lagrange Kovalevskaya -Non-integrable dynamics: Poincaré sections -General principles The PP-torus representation An example Integrable and non-integrable rigid body dynamics

25. Peter Richter Institute for Theoretical Physics 25 Semiclassical quantization of the integrable cases via actions Collect knowledge about non-integrable dynamics; very little seems to be known Consider rigid bodies with Cardan suspension: the configuration space is then not SO(3) but T 3 Outlook

26. Peter Richter Institute for Theoretical Physics 26 Mikhail Kharlamov Igor Gashenenko Alexey Bolsinov Alexander Veselov Holger Dullin Andreas Wittek Dennis Lorek Sven Schmidt Acknowlegdements

28. Peter Richter Institute for Theoretical Physics 28 An example A =(2,1.1,1) s = (0.94868, 0, 0.61623)

29. Peter Richter Institute for Theoretical Physics 29 P 2 h,l S+()S+() S─()S─()