# Action-angle variables for the purely nonlinear oscillator

## TL;DR

This paper analyzes a purely nonlinear oscillator using action-angle variables, deriving its frequency, presenting exact solutions with generalized trigonometric functions, and discussing adiabatic invariance in time-dependent cases.

## Contribution

It introduces a novel application of action-angle variables to a purely nonlinear oscillator, providing exact solutions and insights into adiabatic invariance.

## Key findings

- Frequency matches previous results
- Exact solutions involve generalized trigonometric functions
- Action variable remains invariant under slow parameter changes

## Abstract

In this letter, we study the purely nonlinear oscillator by the method of action-angle variables of Hamiltonian systems. The frequency of the non-isochronous system is obtained, which agrees well with the previously known result. Exact analytic solutions of the system involving generalized trigonometric functions are presented. We also present arguments to show the adiabatic invariance of the action variable for a time-dependent purely nonlinear oscillator.