The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition

TL;DR

This paper proves that under Diophantine conditions, a periodic Hamiltonian system with one degree of freedom can be analytically linearized near an elliptic equilibrium, ensuring stability in the Liapunov sense.

Contribution

It establishes the analyticity of linearization for such systems when the linear part has Diophantine frequencies, extending previous formal linearization results.

Findings

The origin is always stable in the Liapunov sense under the given conditions.

Formal linearizability implies analytic linearizability for the system.

The linearized system's characteristic exponents are purely imaginary with Diophantine frequencies.

Abstract

In this paper we are concerned with the periodic Hamiltonian system with one degree of freedom, where the origin is a trivial solution. We assume that the corresponding linearized system at the origin is elliptic, and the characteristic exponents of the linearized system are $\pm i\omega$ with $\omega$ be a Diophantine number, moreover if the system is formally linearizable, then it is analytically linearizable. As a result, the origin is always stable in the sense of Liapunov in this case.