# The reducibility of quasi-periodic linear Hamiltonian systems and its application to Hill's equation

## TL;DR

This paper proves that under certain conditions, quasi-periodic linear Hamiltonian systems with small perturbations can be simplified to constant systems, and applies this to analyze quasi-periodic Hill's equations.

## Contribution

It establishes reducibility results for quasi-periodic Hamiltonian systems with multiple eigenvalues, extending previous work to more general cases and applying it to Hill's equation.

## Key findings

- Most small perturbations allow reduction to constant systems
- Reduction uses quasi-periodic symplectic transformations
- Application to quasi-periodic Hill's equation demonstrates practical relevance

## Abstract

In this paper, we consider the reducibility of the quasi-periodic linear Hamiltonian system $$\dot{x}=(A+\varepsilon Q(t))x, $$ where $A$ is a constant matrix with possible multiple eigenvalues, $Q(t)$ is analytic quasi-periodic with respect to $t$, and $\varepsilon$ is a sufficiently small parameter. Under some non-resonant conditions, it is proved that, for most sufficiently small $\varepsilon$, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasi-periodic symplectic change of variables with the same basic frequencies as $Q(t)$. Application to quasi-periodic Hill's equation is also given.