1. Peter Richter Institute for Theoretical Physics 1 Iso-Energy Manifolds and Enveloping Surfaces in the Problem of Rigid Body Motions 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 The problems under investigation Of the 4 parameter family of Euler-Poisson problems we address two 2 parameter subsets with arbitrary moments of inertia. The center of gravity is either s = (0,0,0) – Euler family – or s = (1,0,0) – S. B. Katok family. With  = A 2 /A 1 and  = A 3 /A 1 the possible physical values are the colored areas.   2 1 0 1 2 Euler: six equivalent areas, one type of bifurcation diagram Katok: 7 colors corresponding to different bifurcation diagrams Lagrange: white line; Kovalevskaya: red dots

3. Peter Richter Institute for Theoretical Physics 3 Accessible region on Poisson sphere and iso-energy manifolds Euler-Poisson equations construction of iso-energy manifolds Integrable cases Euler Lagrange The Katok family bifurcation diagrams effective potential enveloping surfaces Iso-energy manifolds and enveloping surfaces

4. Peter Richter Institute for Theoretical Physics 4 Euler-Poisson equations coordinates phase space with Poisson structure and 2 Casimirs Hamiltonian equations of motion general constant: energy effective potential on Poisson sphere

5. Peter Richter Institute for Theoretical Physics 5 The basic scheme S2()S2() R3()R3() U 2 h,l E 3 h,l  V 3 h,l P 2 h,l V 3 h,l

6. Peter Richter Institute for Theoretical Physics 6 Euler‘s case l-motion decouples from  -motion S 2 (  ) with effective potential Admissible values in (p,q,r)-space for given (h,l) (h,l) bifurcation diagram 2S 3 S 1 x S 2 RP 3

7. Peter Richter Institute for Theoretical Physics 7 Lagrange‘s case S 1 xS 2 S3S3 RP 3 II:  > 1 I: ½ <  < ¾ RP 3 S3S3 S 1 xS 2 2S 3 RP 3 II: ¾ <  < 1

8. Peter Richter Institute for Theoretical Physics 8 Enveloping surfaces

9. Peter Richter Institute for Theoretical Physics 9 Katok‘s cases 1 2 3 4 56 7 2 3 5 6 7 7 colors for 7 types of bifurcation diagrams 4 S 1 xS 2 7colors for 7 types of energy surfaces 1 2S 3 S3S3 RP 3 K3K3 3S 3 1 2 3 4 5 6 7

10. Peter Richter Institute for Theoretical Physics 10 Effective potentials for case 1 (A 1,A 2,A 3 ) = (1.7,0.9,0.86) l = 2.0 l = 0 l = 1.68 l = 1.71 l = 1.74 l = 1.763 l = 1.773 l = 1.86

11. Peter Richter Institute for Theoretical Physics 11 7+1 types of envelopes (2,1.8)III S 1 xS 2 M32M32 (h,l) = (1,1) S3S3 T2T2 I S3S3 T2T2 (1,0.6) I‘ (2.5,2.15) 2S 3 2T 2 II

12. Peter Richter Institute for Theoretical Physics 12 7+1 types of envelopes (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 (1.9,1.759) VI 3S 3 2S 2, T 2

13. Peter Richter Institute for Theoretical Physics 13 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)