Linear systems with a quadratic integral. (English. Russian original) Zbl 0792.70014

J. Appl. Math. Mech. 56, No. 6, 803-809 (1992); translation from Prikl. Mat. Mekh. 56, No. 6, 900-906 (1992).
Summary: It is shown that a linear system of \(n\) differential equations with constant coefficients, at least one of whose integrals is a non- degenerate quadratic form, may be reduced to a canonical system of Hamiltonian equations. In particular, \(n\) is even and the phase flow preserves the standard measure; if the index of the quadratic integral is odd, the trivial solution is unstable, and so on. For the case \(n=4\) the stability conditions are given in a geometrical form. The general results are used to investigate small oscillations of non-holonomic systems, and also the problem of the stability of invariant manifolds of nonlinear systems that have Morse functions as integrals.


70J25 Stability for problems in linear vibration theory
70F25 Nonholonomic systems related to the dynamics of a system of particles
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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