Hamilton-Jacobi approach for power-law potentials

The classical and relativistic Hamilton-Jacobi approach is applied to the one-dimensional homogeneous potential, V(q) = alphaq 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.

Bibliographic Details

Main Authors:	Santos,R. C., Santos,J., Lima,J. A. S.
Format:	Digital revista
Language:	English
Published:	Sociedade Brasileira de Física 2006
Online Access:	http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332006000700024
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