1995/01/15 by Joseph Katz, Katz, Joseph, Shogo Inagaki +3
Biochemistry, Genetics and Molecular Biology · Earth and Planetary Sciences · Physics and Astronomy · #Astro and Planetary Science #Astrophysics (astro-ph) #FOS: Physical sciences #Geomagnetism and Paleomagnetism Studies #Geophysics and Gravity Measurements #astro-ph
paper · pdf · doi:10.48550/arxiv.astro-ph/9501048
arxiv created 1995/01/15 · openalex publication_date 1995/01/15 · arxiv updated 2009/12/01 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28
The following principle of minimum energy may be a powerful substitute to the\ndynamical perturbation method, when the latter is hard to apply. Fluid elements\nof self-gravitating barotropic flows, whose vortex lines extend to the boundary\nof the fluid, are labelled in such a way that any change of trial\nconfigurations automatically preserves mass and circulation. The velocity field\nis given by a mass conserving Clebsch representation. With three independent\nLagrangian functions, the total energy is stationary for all small variations\nabout a flow with fixed linear and angular momenta provided Euler's equations\nfor steady motion are satisfied. Thus, steady flows are stable if their energy\nis minimum. Since energy is here minimized subject to having local and global\ncontants of the motion fixed, stability limits obtained that way are expected\nto be close to limits given by dynamical perturbation methods. Moreover, the\nstability limits are with respect to arbitrary, not necessary small,\nperturbations. A weaker form of the energy principle is also given which may be\neasier to apply. The Lagrangian functional, with the same three Lagrange\nvariables is stationary for the fully time dependent Euler equations. It\nfollows that the principle of minimum energy gives stability conditions that\nare both necessary and sufficient if terms linear in time derivatives\n(gyroscopic terms) are absent from the Lagrangian. The gyroscopic term for\nsmall deviations around steady flows is given explicitly. Key words: Energy\nvariational principle; Self-gravitating systems; Stability of fluids.\n