Kolmogorov algorithm for isochronous Hamiltonian systems

TL;DR

This paper introduces a Kolmogorov-like algorithm to compute normal forms near invariant tori in isochronous Hamiltonian systems, extending classical methods for coupled oscillators.

Contribution

It develops a novel algorithm for normal form computation in isochronous Hamiltonian systems, bridging Lindstedt and Kolmogorov approaches.

Findings

Algorithm effectively computes normal forms near invariant tori.

Method applicable to systems with polynomial Hamiltonian perturbations.

Provides insights into fixing frequencies in normal form schemes.

Abstract

We present a Kolmogorov-like algorithm for the computation of a normal form in the neighborhood of an invariant torus in `isochronous' Hamiltonian systems, i.e., systems with Hamiltonians of the form $\mathcal{H}=\mathcal{H}_0+\varepsilon \mathcal{H}_1$ where $\mathcal{H}_0$ is the Hamiltonian of $N$ linear oscillators, and $\mathcal{H}_1$ is expandable as a polynomial series in the oscillators' canonical variables. This method can be regarded as a normal form analogue of a corresponding Lindstedt method for coupled oscillators. We comment on the possible use of the Lindstedt method itself under two distinct schemes, i.e., one producing series analogous to those of the Birkhoff normal form scheme, and another, analogous to the Kolomogorov normal form scheme in which we fix in advance the frequency of the torus.