2. Maribor, December 10, 2010 2 Stability of relative equilibria in spinning tops 9th Christmas Symposium dedicated to Prof. Siegfried Großmann CAMTP, University of Maribor December 10, 2010 Peter H. Richter University of Bremen

3. Maribor, December 10, 2010 3 Outline Rigid body dynamics - -parameter space - -reduced phase space Relative equilibria: Staude solutions - -bifurcation diagrams Stability: Grammel analysis Results and Todo Thanks to my student Andreas Krut

4. Maribor, December 10, 2010 4 Rigid body dynamics 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 planar linear 3 principal axes with respect to fixed point center of gravity anywhere relative to that point

5. Maribor, December 10, 2010 5 Euler-Poisson equations coordinates angular velocity angular momentum Casimir constants energy constant → four-dimensional reduced phase space with parameter l

6. Maribor, December 10, 2010 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 possible only for certain combinations of (h, l ): bifurcation diagram

7. Maribor, December 10, 2010 7 Typical bifurcation diagram A = (1.0,1.5, 2.0) s = (0.8, 0.4, 0.3) l h  h stability?

8. Maribor, December 10, 2010 8 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

9. Maribor, December 10, 2010 9 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

10. Maribor, December 10, 2010 10 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

11. Maribor, December 10, 2010 11 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 

12. Maribor, December 10, 2010 12 Same center of gravity, but permutation of moments of inertia

13. Maribor, December 10, 2010 13 M

14. Maribor, December 10, 2010 14

15. Maribor, December 10, 2010 15 Results The stability analysis is surprisingly simple, but the complexity of its results exceeds that of the bifurcation diagrams. Todo A number of typical scenarios have been identified, and Andreas Krut has written a powerful Matlab program. The complete parameter dependence in the 4D set of moments of inertia and center of gravity locations seems within reach and should be established. In addition to the eigenvalues, the eigensolutions of the variational equations should also be determined. But keep in mind: Relative equilibria are only the simplest aspect of rigid body dynamics

17. Maribor, December 10, 2010 17

18. Maribor, December 10, 2010 18 Siegfried Großmann 1990