Scaling Symmetry Makes Spherical Hydrostatics Integrable

Details Scaling Symmetry Makes Spherical Hydrostatics Integrable Scaling Symmetry Makes Spherical Hydrostatics Integrable

2011-12-19

arxiv.org/abs/1112.4223v4

Physics

Mathematical Physics

12 pages, 4 figures, 3 tables. arXiv admin note: substantial text overlap with arXiv:1106.1222

Scientific paper

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.

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