The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body

Abstract

Contents Introduction §1. The general Maupertuis principle §2. The Maupertuis principle in the dynamics of a massive rigid body §3. The Maupertuis principle and the explicit form of the metric generated on the sphere by a quadratic Hamiltonian on the Lie algebra of the group of motions of R3 §4. Classical cases of integrability in rigid body dynamics and the corresponding geodesic flows on the sphere §5. Integrable metrics on the torus and on the sphere §6. Conjectures §7. The complexity of integrable geodesic flows of on the sphere and on the torus §8. A rougher conjecture: the complexities of non-singularly integrable metrics on the sphere or on the torus coincide with those of the known integrable §9. The geodesic flow on an ellipsoid is topologically orbitally equivalent to the Euler integral case in the dynamics of a rigid body Bibliography