28,395 research outputs found
Hamilton-Jacobi Approach for Power-Law Potentials
The classical and relativistic Hamilton-Jacobi approach is applied to the
one-dimensional homogeneous potential, , where and
are continuously varying parameters. In the non-relativistic case, the
exact analytical solution is determined in terms of , and the total
energy . It is also shown that the non-linear equation of motion can be
linearized by constructing a hypergeometric differential equation for the
inverse problem . 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 , it leads to a simple harmonic oscillator if , an
"anti-oscillator" if , 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
. For , it is found that the correction is just twice that one
deduced for the simple harmonic oscillator (), and does not depend on the
specific value of .Comment: 12 pages, Late
Jeans' gravitational instability and nonextensive kinetic theory
The concept of Jeans gravitational instability is rediscussed in the
framework of nonextensive statistics and its associated kinetic theory. A
simple analytical formula generalizing the Jeans criterion is derived by
assuming that the unperturbed self- gravitating collisionless gas is
kinetically described by the -parameterized class of power law velocity
distributions. It is found that the critical values of wavelength and mass
depend explicitly on the nonextensive -parameter. The standard Jeans
wavelength derived for a Maxwellian distribution is recovered in the limiting
case =1. For power-law distributions with cutoff, the instability condition
is weakened with the system becoming unstable even for wavelengths of the
disturbance smaller than the standard Jeans length .Comment: 5 pages, including 3 figures. Accepted for publication in A&
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