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 directly 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
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