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

2. Jacobs University Feb. 23, 2011 2 Outline Rigid bodies: configuration and parameter spaces - -SO(3)→S 2, T 3 →T 2 - -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 T 3 -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

3. Jacobs University Feb. 23, 2011 3 Rigid bodies in SO(3) two moments of inertia  two angles  for the center of gravity s 1, s 2, s 3 4 essential parameters after scaling of lengths, time, energy: One point fixed in space, the rest free to move 3 principal axes with respect to fixed point center of gravity anywhere relative to that point planar linear LagrangeGeneral Euler

4. Jacobs University Feb. 23, 2011 4 Rigid bodies in T 3 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: Cardan angles (     )  a little more than 2 SO(3) → classical spin? Lagrange: up – Integr horiz – Chaos Euler: symm up – Integr tilted – Chaos General: horiz – Interm asymm up – Chaos

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

6. 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

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

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

9. Jacobs University Feb. 23, 2011 9 Euler‘s case  -motion decouples from  -motion Poisson sphere potential(h,l)-bifurcation diagram B iso-energy surfaces in reduced phase space: , S 3, S 1 xS 2, RP 3 foliation by 1D invariant tori S3S3 S 1 xS 2 RP 3 

10. Jacobs University Feb. 23, 2011 10 Lagrange‘s case ¾ <  < 1 RP 3 S3S3 2S 3 S 1 xS 2 cigar:   S 1 xS 2 S3S3 RP 3 disk: ½ <  < ¾ Poisson sphere potentials B

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

12. Jacobs University Feb. 23, 2011 12 EulerLagrangeKovalevskaya Energy surfaces in action representation B

13. Jacobs University Feb. 23, 2011 13 Poincaré section E 3 h,l P 2 h,l U 2 h,l V 2 h,l S2()S2() R3()R3() Poisson sphere accessible velocities S = 0

14. Jacobs University Feb. 23, 2011 14 Topology of Surface of Section if l z 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 T 3 -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 l z P S

15. Jacobs University Feb. 23, 2011 15 Examples  (s 1,s 2,s 3 ) = (1,0,0)  (s 1,s 2,s 3 ) = (1,0,0) integrable non-integrable 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

16. 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 T 3, 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 T 3 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

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

19. Jacobs University Feb. 23, 2011 19 Stability analysis: variational equations (Grammel 1920) relative equilibrium: variation: variational equations: J: a 6x6 matrix with rank 4 and characteristic polynomial g 0 6 + g 1 4 + g 2 2

20. Jacobs University Feb. 23, 2011 20 Stability analysis: eigenvalues 2 eigenvalues  = 0 4 eigenvalues obtained from g 0 4 + g 1 2 + g 2  The two 2 are either real or complex conjugate. If the 2 form a complex pair, two have positive real part → instability If one 2 is positive, then one of its roots is positive → instability Linear stability requires both solutions 2 to be negative: then all are imaginary We distinguish singly and doubly unstable branches of the bifurcation diagram depending on whether one or two 2 are non-negative

21. 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

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

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

24. Jacobs University Feb. 23, 2011 24 M

25. Jacobs University Feb. 23, 2011 25