MOMENTUM MAPS, INDEPENDENT FIRST INTEGRALS AND INTEGRABILITY FOR CENTRAL POTENTIALS

Abstract

It is shown that all the motions of a natural Hamiltonian H(q, p) = ½‖p‖2+V(q) lie on planes through 0 ∈ Rn if and only if V is a central potential, i.e. H admits SO(n) symmetry. Then, using the momentum maps associated to their natural symmetry groups, we study in detail the functional independence of first integrals of a general central potential, of the isotropic harmonic potential and of the Kepler potential. We show that all the smooth first integral of the isotropic harmonic oscillator are dependent of the angular momentum tensor L and of the Fradkin tensor H, and that all the smooth first integrals of the Kepler system on the region of negative energy are dependent of the angular momentum tensor L and of the Laplace–Runge–Lenz vector B.