Scaling Symmetry and Integrable Spherical Hydrostatics

• 作者: Dallas C. Kennedy Sidney Bludman
• 来源期刊: 现代物理(英文)
• 学科: 数学
• 关键词: LAGRANGIAN Mechanics SYMMETRY Hydrodynamics ASTROPHYSICS STELLAR Structure
• 年，卷（期）: 2013,（4）
• 页码范围: 486-494
• 页数: 9页
• 分类号: O1

摘要

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 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 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.