Vektorfeldalgebren, Gruppenaktionen und das Keplerproblem

Abstract

AbstractWe supply an analytic proof of the theorem of Palais about the existence of a Lie group action on a compact manifold M with a Lie algebra 𝔳 of vector fields. Every compact connected integral manifold N of 𝔳 is then diffeomorphic to a homogeneous manifold. In the special case of a Lie algebra of Hamiltonian vector fields, the group action consists of symplectic diffeomorphisms. Applications are the non‐commutative version of the Cauchy initial value problem for Hamilton‐Jacobi systems and of Liouville's theorem. As examples with non‐compact M, we determine group actions for the harmonic oscillator and for Kepler's problem.