Some tensor invariants of geodesic, potential, and dissipative systems with four degrees of freedom

M. V. Shamolin

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 190-211.

Résumé

The detection of a sufficient number of tensor invariants (and not only the first integrals), as [13, 14, 45] is known, allows integrating a system of differential equations. For example, the presence of an invariant differential form of the phase volume makes it possible to reduce the number of required first integrals. As you know, this fact is natural for conservative systems. For systems with attracting or repelling limit sets, not only some first integrals, but also the coefficients of the available invariant differential forms should, generally speaking, include transcendental (i.e. having essentially singular points, in the sense of complex analysis) functions (see also [1, 23, 24]). We briefly give examples of frequently occurring tensor invariants. Scalar invariants are the first integrals of the system under consideration. Invariant vector fields are symmetry fields for a given system (they commute with the vector field of the system under consideration). The phase flows of systems of differential equations generated by these fields translate the solutions of the system in question into solutions of the same system. Invariant external differential forms (which is mainly carried out in this paper) generate integral invariants of the system under consideration. At the same time, the vector field of the system under consideration itself is one of the invariants (a trivial invariant). Knowledge of tensor invariants of the system of differential equations under consideration facilitates both its integration and qualitative research. Our approach consists in the fact that in order to accurately integrate an autonomous system of $n$ differential equations, in addition to the mentioned trivial invariant, it is necessary to know $n-1$ independent tensor invariants. In this paper, we present tensor invariants (differential forms) for homogeneous dynamical systems on the tangent bundles of smooth four-dimensional manifolds and demonstrate the connection between the availability of these invariants and the existence of a complete set of first integrals, which is necessary for integrating of geodesic, potential, and dissipative systems.

Détail

Keywords: dynamical system, integrability, dissipation, transcendental first integral, invariant differential form.

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