  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.
