Hamilton-Jacobi Approach for Power-Law Potentials

The classical and relativistic Hamilton-Jacobi approach is applied to the one-dimensional homogeneous potential, V(q) = aqn, where a and n are continuously varying parameters. In the non-relativistic case, the exact analytical solution is determined in terms of a, 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 > 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.