Abstract
New cases of integrable seventh-order dynamical systems that are homogeneous with respect to some of the variables are presented, in which a system on the tangent bundle of a three-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external component, which has dissipation of different signs. The external field is introduced using some unimodular transformation and generalizes previously considered fields. Complete sets of both first integrals and invariant differential forms are given.
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REFERENCES
H. Poincaré, Calcul des probabilités (Gauthier-Villars, Paris, 1912).
A. N. Kolmogorov, “On dynamical systems with an integral invariant on the torus,” Dokl. Akad. Nauk SSSR 93 (5), 763–766 (1953).
V. V. Kozlov, “Tensor invariants and integration of differential equations,” Russ. Math. Surv. 74 (1), 111–140 (2019).
M. V. Shamolin, “On integrability in transcendental functions,” Russ. Math. Surv. 53 (3), 637–638 (1998).
M. V. Shamolin, “Complete list of first integrals of dynamic equations of motion of a 4D rigid body in a nonconservative field under the assumption of linear damping,” Dokl. Phys. 58 (4), 143–146 (2013).
M. V. Shamolin, “Invariants of fifth-order homogeneous systems with dissipation,” Dokl. Math. 108 (3), 506–513 (2023).
M. V. Shamolin, “Invariant volume forms of variable dissipation systems with three degrees of freedom,” Dokl. Math. 106 (3), 479–484 (2022).
V. V. Kozlov, “Rational integrals of quasi-homogeneous dynamical systems,” J. Appl. Math. Mech. 79 (3), 209–216 (2015).
F. Klein, Vorlesungen über nicht-euklidische Geometrie (VDM, Müller, Saarbrücken, 2006).
H. Weyl, Symmetry (Princeton Univ. Press, Princeton, N.J., 2016).
V. V. Kozlov, “Integrability and non-integrability in Hamiltonian mechanics,” Russ. Math. Surv. 38 (1), 1–76 (1983).
V. V. Trofimov and M. V. Shamolin, “Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems,” J. Math. Sci. 180 (4), 365–530 (2012).
M. V. Shamolin, “New cases of full integrability in dynamics of a dynamically symmetric four-dimensional solid in a nonconservative field,” Dokl. Phys. 54 (3), 155–159 (2009).
M. V. Shamolin, “Complete list of first integrals in the problem on the motion of a 4D solid in a resisting medium under assumption of linear damping,” Dokl. Phys. 56 (9), 498–501 (2011).
E. Kamke, Gewohnliche Differentialgleichungen, 5th ed. (Akademie-Verlag, Leipzig, 1959).
A. D. Polyanin and V. F. Zaitsev, Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, 3rd ed. (Chapman and Hall/CRC, New York, 2017). https://doi.org/10.1201/9781315117638
B. V. Shabat, Introduction to Complex Analysis (Nauka, Moscow, 1987; Am. Math. Soc. Providence, R.I., 1992).
S. P. Novikov and I. A. Taimanov, Modern Geometric Structures and Fields (Mosk. Tsentr Neprer. Mat. Obrazovan., Moscow, 2005; Am. Math. Soc., Providence, R.I., 2006).
I. Tamura, Topology of Foliations: An Introduction (Am. Math. Soc., Providence, R.I., 2006).
M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications,” J. Math. Sci. 162 (6), 741–908 (2009).
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Translated by I. Ruzanova
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Shamolin, M.V. Invariants of Seventh-Order Homogeneous Dynamical Systems with Dissipation. Dokl. Math. 109, 152–160 (2024). https://doi.org/10.1134/S1064562424701941
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DOI: https://doi.org/10.1134/S1064562424701941