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Hamilton-Jacobi approach for power-law potentials

DOI: 10.1590/S0103-97332006000700024

Keywords: hamilton-jacobi equation, power-law potentials.

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Abstract:

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 < 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.

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