Hamilton-Jacobi Approach for Power-Law Potentials

TL;DR

This paper applies the Hamilton-Jacobi method to analyze power-law potentials in both classical and relativistic contexts, deriving exact solutions, transformations, and period corrections for various energy regimes and potential exponents.

Contribution

It provides a comprehensive analytical framework for solving power-law potentials using Hamilton-Jacobi theory, including linearization techniques and relativistic corrections.

Findings

Exact solutions for non-relativistic power-law potentials.

Transformation to harmonic oscillator form for E>0, anti-oscillator for E<0, free particle for E=0.

First order relativistic period correction, especially for large n.

Abstract

The classical and relativistic Hamilton-Jacobi approach is applied to the one-dimensional homogeneous potential, $V(q)=\alpha q^n$, where $\alpha$ and $n$ are continuously varying parameters. In the non-relativistic case, the exact analytical solution is determined in terms of $\alpha$, $n$ and the total energy $E$. It is also shown that the non-linear equation of motion can be linearized by constructing a hypergeometric differential equation for the inverse problem $t(q)$. A variable transformation reducing the general problem to that one of a particle subjected to a linear force is also established. For any value of $n$, it leads to a simple harmonic oscillator if $E>0$, an "anti-oscillator" if $E<0$, or a free particle if E=0. However, such a reduction is not possible in the relativistic case. For a bounded relativistic motion, the first order correction to the period is determined for any value of $n$. For $n >> 1$, it is found that the correction is just twice that one deduced for the simple harmonic oscillator ($n=2$), and does not depend on the specific value of $n$.