2011/06/07 by S. A. Bludman, Sidney Bludman, Bludman, Sidney · 1 citation
Mathematics · Physics and Astronomy · #Astro and Planetary Science #Stellar, planetary, and galactic studies #astro-ph.SR #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.48550/arxiv.1106.1222
10 pages, 4 figures
arxiv created 2011/06/07 · arxiv updated 2011/06/08
Any symmetry reduces a second-order differential equation to a first-order equation: 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 \em non-conservation laws between invariants. We obtain these 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 published properties of polytropes and define a \em core radius, a new measure of mass concentration in polytropes of index n. 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, emphasizing how the same mechanical structure allows different \em thermostatic structures in relativistic degenerate white dwarfs and and zero age main sequence stars.