Scaling Symmetry and Integrable Spherical Hydrostatics
  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 {\em non-conservation laws} between invariants. We obtain these non-conservation laws by extending Noether's Theorem to non-variational symmetries and present an innovative variational formulation of spherical adiabatic hydrostatics. For the scale-invariant case, this novel synthesis of group theory, hydrostatics, and astrophysics allows us to recover all the known properties of polytropes and define a {\em 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 obtained, along with useful approximations. An appendix discusses the $n=3$ polytrope in order to emphasize how the same mechanical structure allows different thermal structures in relativistic degenerate white dwarfs and zero age main sequence stars. 
Sidney Bludman, Dallas C. Kennedy
math-ph astro-ph.SR math.MP physics.class-ph physics.flu-dyn