Hamilton-Jacobi approach for power-law potentials

Author(s)

R. C. Santos ^[1] ; J. Santos ^[2] ; J. A. S. Lima ^[3]

Affiliation

^[1] UFRN. Departamento de Física - Brasil

^[2] UFRN. Departamento de Física - Brasil

^[3] UFRN. Departamento de Física - Brasil

• Document type: Journal article
• Source: Brazilian Journal of Physics; v. 36, p. 1257-1261, 2006-12-00.

Abstract

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. (AU)

• FAPESP's process: 04/13668-0 - Cosmology in the era of precision: dark energy, dark matter and background cosmic radiation
• Grantee: Jose Ademir Sales de Lima
• Support Opportunities: Research Projects - Thematic Grants