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

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 $V_{2n}(y_{2n})=\frac{1}{2n}{k}_{2n}{y}_{2n}^{2n}(t){|}_{n\ge 1}$). The nonlinear quartic oscillator corresponds to $n=2$ while the harmonic oscillator corresponds to $n=1$.