On stability of equilibria in a~pseudo-Riemannian space

V. V. Kozlov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 80 (2025) no. 1, pp. 51-74

Résumé

The stability of equilibria is considered for systems whose kinetic energy is a pseudo-Riemannian metric on the configuration space. Equilibria are critical points of the potential energy. For a linear system with two degrees of freedom the stability diagram is plotted and the bifurcations of eigenvalues are indicated. Points of maximum and minimum of the potential energy are unstable equilibria in the pseudo-Euclidean case. The same conclusion holds for nonlinear analytic systems with two degrees of freedom. Conditions for stability are indicated for multidimensional linear systems in a pseudo-Euclidean space. In particular, an equilibrium is stable if and only if the linear equations of motion can be reduced to a ‘natural’ system with positive definite kinetic energy and, in addition, the potential energy takes a strict minimum at this equilibrium. The influence of dissipative and gyroscopic forces on the stability of equilibria in a pseudo-Riemannian space is investigated. The instability of an isolated equilibrium is proved in the case when dissipative forces with full energy dissipation are added. The instability degree is calculated for linear dissipative systems. Conditions for the stability of linear systems in the case when large gyroscopic forces are applied to them are indicated. Bibliography: 40 titles.

Détail

Keywords: pseudo-Riemannian metric, critical points, Largange–Dirichlet theorem, pencils of quadratic forms, Brunella's theorem, resonances, asymptotic trajectories, instability degree, dissipative and gyroscopic forces, Thomson–Tait theorem, formal stability.
Mots-clés : Lagrange's equations, inertia indices