A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows

TL;DR

This paper introduces a simple, globally applicable method for deriving second-order normal forms of perturbed Hamiltonian systems relative to their periodic flows, demonstrated through classical examples.

Contribution

It presents a novel, easy-to-implement formalism for second-order normal forms in Hamiltonian systems that is applicable globally and demonstrated on well-known models.

Findings

The formalism is globally applicable and easy to implement.

It effectively derives second-order normal forms for specific Hamiltonian systems.

The method is illustrated with the Hénon-Heiles and elastic pendulum examples.

Abstract

We study the determination of the second-order normal form for perturbed Hamiltonians $H_{\epsilon}=H_0 +\epsilon H_1 +\frac{\epsilon^2}{2} H_2$, relative to the periodic flow of the unperturbed Hamiltonian $H_0$. The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the H\'enon-Heiles and the elastic pendulum Hamiltonians.