On Lagrangian methods for the study of small perturbations in stellar systems

Details On Lagrangian methods for the study of small perturbations in stellar systems On Lagrangian methods for the study of small perturbations in stellar systems

Jun 1989

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989apj...341..105v&link_type=abstract

Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 341, June 1, 1989, p. 105-112.

Mathematics

11

Method Of Characteristics, Perturbation Theory, Stellar Oscillations, Equations Of Motion, Hermitian Polynomial, Hilbert Space, Lagrange Coordinates, Matrices (Mathematics)

Scientific paper

The characteristic value problem is formulated in a Lagrangian representation for the normal modes of oscillation of a stellar system. The problem is shown to be non-Hermitian, and the proper adjoint characteristic value problem is identified. A transformation is found which turns the characteristic vector for a given frequency into the adjoint characteristic vector for another frequency and permits the elimination of the adjoint vectors from all calculations. A variational principle is constructed for the solution of the general characteristic value problem. For systems that are invariant with respect to the simultaneous application of time reversal and reflection through a given plane, a symmetry operator is constructed which turns the characteristic vector for a given frequency into the characteristic vector for the complex-conjugate frequency, a transformation is found which turns each characteristic vector into its adjoint characteristic vector, and a simpler version of the variational principle is constructed. These variational principles provide a basis for approximate methods for the solution of the characteristic value problem. The results presented here have well known counterparts in fluid dynamics.

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