The complex dynamics of spinning tops

1. The complex dynamics of spinning tops Physics Colloquium Jacobs University Bremen February 23, 2011 Peter H. RichterUniversity of Bremen Jacobs University Feb. 23, 2011

2. Outline Rigid bodies: configuration and parameter spaces • SO(3)→S2, T3→T2 • Moments of inertia, center of gravity, Cardan frame SO(3)-Dynamics • Euler-Poisson equations, Casimir and energy constants • Relative equilibria (Staude solutions) and their stability (Grammel) • Bifurcation diagrams, iso-energy surfaces • Integrable cases: Euler, Lagrange, Kovalevskaya • Liouville-Arnold foliation, critical tori, action representation • General motion: Poincaré section over Poisson-spheres→torus T3-Dynamics • canonical equations • 3D or 5D iso-energy surfaces • Integrable cases: symmetric Euler and Lagrange in upright Cardan frame • General motion: Poincaré section over Poisson-tori+2cylinder connection Jacobs University Feb. 23, 2011

3. planar linear linear planar planar linear Rigid bodies in SO(3) One point fixed in space, the rest free to move 3 principal axes with respect to fixed pointcenter of gravity anywhere relative to that point 4 essential parameters after scaling of lengths, time, energy: two moments of inertiaa, b (g = 1- a- b) two angles s,t for the center of gravity s1, s2, s3 Euler Lagrange General Jacobs University Feb. 23, 2011

4. Cardan angles (j,q, y) (j + p, 2p - q, y + p) Rigid bodies in T3 a little more than 2 SO(3) → classical spin? 6 essential parameters after scaling of lengths, time, energy: Euler: symm up – Integr two moments of inertiaa, b (g = 1-a-b) asymm up – Chaos at least one independent moment of inertia r for the Cardan frame Lagrange: up – Integr two angles s,t for the center of gravity tilted – Chaos angledbetween the frame‘s axis and the direction of gravity General: horiz – Interm horiz – Chaos Jacobs University Feb. 23, 2011

5. coordinates angular velocity angularmomentum Casimir constants energy constant SO(3)-Dynamics: Euler-Poisson equations → four-dimensional reduced phase space with parameter l Jacobs University Feb. 23, 2011

6. Relative equilibria: Staude solutions angular velocity vector constant, aligned with gravity high energy: rotations about principal axes low energy: rotations with hanging or upright position of center of gravity intermediate energy: carrousel motion possible only for certain combinations of (h,l ): bifurcation diagram Jacobs University Feb. 23, 2011

7. l l2 h h stability? Typical bifurcation diagram A = (1.0,1.5, 2.0) s = (0.8, 0.4, 0.3) Jacobs University Feb. 23, 2011

8. A Integrable cases P Euler: „gravity-free“ E 4 integrals Lagrange: „heavy“, symmetric L 3 integrals Kovalevskaya: K 3 integrals Jacobs University Feb. 23, 2011

9. (h,l)-bifurcation diagram Poisson sphere potential Euler‘s case w-motiondecouples from g-motion  S3 S1xS2 RP3 iso-energy surfaces in reduced phase space: , S3, S1xS2, RP3 foliation by 1D invariant tori B Jacobs University Feb. 23, 2011

10. Lagrange‘s case Poisson sphere potentials disk: ½ < a < ¾ 2S3 cigar:a > 1 S3 ¾ < a < 1 S1xS2 B RP3 RP3 S3 S1xS2 Jacobs University Feb. 23, 2011

11. Kovalevskaya‘s case Tori in phase space and Poincaré surface of section Action integral: B Jacobs University Feb. 23, 2011

12. Energy surfaces in action representation Euler Lagrange Kovalevskaya B Jacobs University Feb. 23, 2011

13. Poincaré section E3h,l S = 0 P2h,l U2h,l V2h,l R3(w) S2(g) Poissonsphere accessible velocities Jacobs University Feb. 23, 2011

14. Topology of Surface of Section if lz is an integral SO(3)-Dynamics • 1:1 projection to 2 copies of the Poisson sphere which are punctuated at their poles and glued along the polar circles • this turns them into a torus (PP torus) • at high energies the SoS covers the entire torus • at lower energies boundary points on the two copies must be identified T3-Dynamics • 1:1 projection to 2 copies of the Poisson torus plus two connecting cylinders • the Poincaré surface is not a manifold! • but it allows for a complete picture at given energy h and angular momentum lz P S Jacobs University Feb. 23, 2011

15. Examples non-integrable integrable (a,b,g) = (0.4, 0.4, 0.2) (s1,s2,s3) = (1,0,0) (a,b,g) = (0.49, 0.27, 0.24) (s1,s2,s3) = (1,0,0) black: in dark: out light: – black: out dark: in light: – black: in dark: out light: – black: out dark: in light: – In both cases is the surface of section a torus: part of the PP torus, outermost circles glued together B Jacobs University Feb. 23, 2011

16. Summary • Rigid bodies fixed in one point and subject to external forces need a support, e. g. a Cardan suspension • This changes the configuration space from SO(3) to T3, and the parameter set from 4 to 6 dimensional • Integrable cases are only a small albeit highly interesting subset • Not much is known about non-integrable cases • If one degree of freedom is cyclic, complete Poincaré surfaces of section can be identified – always with SO(3), sometimes with T3 • The general case with 3 non-reducible degrees of freedom is beyond currently available methods of investigation • Very little is known about the quantum mechanics of such systems Jacobs University Feb. 23, 2011

17. Thanks to • Emil Horozov • Mikhail Kharlamov • Igor Gashenenko • Alexey Bolsinov • Alexander Veselov • Victor Enolskii • Nadia Juhnke • Andreas Wittek • Holger Dullin • Sven Schmidt • Dennis Lorek • Konstantin Finke • Nils Keller • Andreas Krut Jacobs University Feb. 23, 2011

19. relative equilibrium: variation: variational equations: Stability analysis: variational equations (Grammel 1920) J: a 6x6 matrix with rank 4 and characteristic polynomial g0l6 + g1l4 + g2l2 Jacobs University Feb. 23, 2011

20. Stability analysis: eigenvalues 2 eigenvalues l = 0 4 eigenvalues obtained fromg0l4 + g1l2 + g2 = 0 The two l2 are either real or complex conjugate. If the l2 form a complex pair, two l have positive real part → instability If one l2 is positive, then one of its roots l is positive → instability Linear stability requires both solutions l2 to be negative: then all l are imaginary We distinguish singly and doubly unstable branches of the bifurcation diagram depending on whether one or two l2 are non-negative Jacobs University Feb. 23, 2011

21. Typical scenario • hanging top starts with two pendulum motions and develops into rotation about axis with highest moment of inertia (yellow) • upright top starts with two unstable modes, then develops oscillatory behaviour and finally becomes doubly stable (blue) • 2 carrousel motions appear in saddle node bifurcations, each with one stable and one singly unstable branch. The stable branches join with the rotations about axes of largest (red) and smallest (green) moments of inertia. The unstable branches join each other and the unstable Euler rotation Jacobs University Feb. 23, 2011

22. g3 g1 g2 stable hanging rotation about 1-axis (yellow) connects to upright carrousel motion (red) unstable carrousel motion about 2-axis (red and green) connects to stable branches stable upright rotation about 3-axis (blue) connects to hanging carrousel motion (green) w Orientation of axes, and angular velocities Jacobs University Feb. 23, 2011

23. Same center of gravity, but permutation of moments of inertia Jacobs University Feb. 23, 2011

24. M Jacobs University Feb. 23, 2011

25. Jacobs University Feb. 23, 2011