Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. III. Force fields with dissipation

M. V. Shamolin

Geodesic

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Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. III. Force fields with dissipation

M. V. Shamolin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 212 (2022), pp. 120-138.

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Résumé

In many problems of dynamics, systems arise whose position spaces are four-dimensional manifolds. Naturally, the phase spaces of such systems are the tangent bundles of the corresponding manifolds. Dynamical systems considered have variable dissipation, and the complete list of first integrals consists of transcendental functions expressed in terms of finite combinations of elementary functions. In this paper, we prove the integrability of more general classes of homogeneous dynamical systems with variable dissipation on tangent bundles of four-dimensional manifolds. The first part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. I. Equations of geodesic lines// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 210 (2022), pp. 77–95. The second part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. II. Potential force fields// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 211 (2022), pp. 29–40.

Keywords: dynamical system, nonconservative field, integrability, transcendental first integral.

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     author = {M. V. Shamolin},
     title = {Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. {III.} {Force} fields with dissipation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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M. V. Shamolin. Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. III. Force fields with dissipation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 212 (2022), pp. 120-138. http://geodesic.mathdoc.fr/item/INTO_2022_212_a12/

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