# On the transformations linearizing isochronous centers of Hamiltonian systems

## TL;DR

This paper characterizes when planar Hamiltonian systems with isochronous centers can be globally linearized through analytic transformations, linking the structure of the level curve to the existence of such transformations.

## Contribution

It establishes necessary and sufficient conditions for the existence of global analytic linearizing transformations for isochronous centers in Hamiltonian systems.

## Key findings

- Linearizing transformations exist if and only if the level curve contains only isolated points.
- Such transformations can be extended to the entire plane when the origin is the unique center.
- The results connect the geometric structure of level curves with linearizability of Hamiltonian centers.

## Abstract

In this paper we study the transformations linearizing isochronous centers of planar Hamiltonian differential systems with polynomial Hamiltonian functions $H(x,y)$ having only isolated singularities. Assuming the origin is an isochronous center lying on the level curve $L_0$ defined by $H(x,y)=0$, we prove that, there exists a canonical linearizing transformation analytic on a simply-connected open set $\Omega$ with closure $\overline{\Omega}=\mathbb{R}^2$, if and only if, $L_0$ consists of only isolated points; furthermore, if the origin is the unique center, then the condition that $L_0$ consists of only isolated points implies that the corresponding canonical linearizing transformation can be analytically defined on the whole plane.