On a theorem of Lyapounov

TL;DR

This paper proves the convergence of normal form transformations for Hamiltonian systems near equilibrium under non-resonance conditions, extending Lyapounov's classical results by removing the pure imaginary eigenvalue restriction.

Contribution

It establishes the convergence of the normal form transformation for Hamiltonian systems without requiring eigenvalues to be purely imaginary, generalizing Lyapounov's theorem.

Findings

Normal form transformation converges under non-resonance conditions.

Invariant manifolds are constructed for the Hamiltonian system.

Extension of Lyapounov's result to more general eigenvalues.

Abstract

It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case the eigenvalues of the linearized system satisfy non--resonance conditions of Melnikov's type. The normal form possesses a two dimensional (local) invariant manifold on which the solutions are known. If the eigenvalue is pure imaginary then these solutions are the natural continuation of a normal mode of the linear system. The latter result was first proved by Lyapounov. The present paper completes Lyapounov's result in that the convergence of the transformation of the Hamiltonian to a normal form is proven and the condition that the eigenvalues be pure imaginary is removed.