Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2006-08-30
Braz.J.Phys. 36 (2006) 1257-1261
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
12 pages, Latex
Scientific paper
The classical and relativistic Hamilton-Jacobi approach is applied to the one-dimensional homogeneous potential, $V(q)=\alpha q^n$, 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 $n$.
Lima Jose A. S.
Santos Joedson
Santos R. C.
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