1112.4223 (Sidney Bludman et al.)

Scaling Symmetry Makes Spherical Hydrostatics Integrable    [PDF]

Sidney Bludman, Dallas C. Kennedy

Any symmetry reduces a second-order differential equation to a first
integral: variational symmetries of the action (exemplified by central field
dynamics) lead to conservation laws, but symmetries of only the equations of
motion (exemplified by scale-invariant hydrostatics) yield first-order
non-conservation laws between invariants. We obtain these non-conservation laws
by extending Noether's Theorem to non-variational symmetries and present a
variational formulation of spherical adiabatic hydrostatics. For
scale-invariant hydrostatics, we recover all the known properties of polytropes
and define a core radius, inside which polytropes of index n share a common
core mass density structure, and outside of which their envelopes differ. The
Emden solutions (regular solutions of the Lane-Emden equation) are finally
obtained, along with useful approximations. An appendix discusses the special
n=3 polytrope in order to emphasize how the same mechanical structure allows
different thermostatic structures in relativistic degenerate white dwarfs and
zero age main sequence stars.

View original: http://arxiv.org/abs/1112.4223