Actions for an Hierarchy of Attractive Nonlinear Oscillators Including the Quartic Oscillator in 1+1 Dimensions

TL;DR

This paper derives explicit formulas for the classical action of a hierarchy of nonlinear oscillators with even power potentials, including the quartic oscillator, using Hamilton-Jacobi theory.

Contribution

It provides explicit end-point data formulas for the classical action of nonlinear oscillators with even power potentials, extending to the quartic oscillator.

Findings

Explicit classical action formulas for nonlinear oscillators.

Hierarchy includes harmonic and quartic oscillators.

Application of Hamilton-Jacobi equation to these systems.

Abstract

In this paper, we present an explicit form in terms of end-point data for the classical action $S_{2n}$ evaluated on extremals satisfying the Hamilton-Jacobi equation for each member of a hierarchy of classical non-relativistic oscillators characterized by even power potentials (i.e., attractive potentials $V_{2n}(y_{2n})={\frac{1}{2n}}k_{2n}y_{2n}^{2n}(t)|_{n{\geq}1}$). The nonlinear quartic oscillator corresponds to $n=2$ while the harmonic oscillator corresponds to $n=1$.