The Emden solutions (regular solutions of the Lane-Emden equation) are obtained, along with useful approximations.

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 core radius, inside which polytropes of index n share a common core mass density structure, and outside of which their envelopes differ. 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. Lagrangian Mechanics Symmetry Hydrodynamics Astrophysics Stellar StructureĪny 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.

Scaling Symmetry and Integrable Spherical HydrostaticsĪUTHORS: Sidney Bludman, Dallas C. DiPrima, “Elementary Differential Equations and Boundary Value Problems,” 7th Edition, John Wiley and Sons, Hoboken, 2001.

Boyce e diprima