2012/12/02 by S. A. Bludman, Sidney Bludman, Andrés E. Guzmán +5
Chemistry · Mathematics · Physics and Astronomy · #Advanced Physical and Chemical Molecular Interactions #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1212.0179
7 pages, 6 figures
arxiv created 2012/12/02 · openalex publication_date 2012/12/02 · arxiv updated 2012/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
While not generally a conservation law, any symmetry of the equations of motion implies a useful reduction of any second-order equationto a first-order equation between invariants, whose solutions (first integrals) can then be integrated by quadrature (Lie's Theorem on the solvability of differential equations). We illustrate this theorem by applying scale invariance to the equations for the hydrostatic equilibrium of stars in local thermodynamic equilibrium: Scaling symmetry reduces the Lane-Emden equation to a first-order equation between scale invariants un; vn, whose phase diagram encapsulates all the properties of index-n polytropes. From this reduced equation, we obtain the regular (Emden) solutions and demonstrate graphically how they transform under scale transformations.