
HamiltonJacobi approach for powerlaw potentialsDOI: 10.1590/S010397332006000700024 Keywords: hamiltonjacobi equation, powerlaw potentials. Abstract: the classical and relativistic hamiltonjacobi approach is applied to the onedimensional homogeneous potential, v(q) = aqn, where a and n are continuously varying parameters. in the nonrelativistic case, the exact analytical solution is determined in terms of a, n and the total energy e. it is also shown that the nonlinear 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 "antioscillator" 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.
