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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.

MSC:

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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References:

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