  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.
