Abstract
In this paper, we prove the integrability of some classes of odd-order dynamical systems (namely, systems of order 3, 5, and 7), which are homogeneous in some variables and contain a system on the tangent bundle of a smooth manifolds. In this case, we separate force fields into internal (conservative) and external, which has sign-alternating dissipation. External fields are introduced by using some unimodular transformations and generalize fields considered earlier.
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References
N. Bourbaki, Lie Groups and Algebras [Russian translation], Mir, Moscow (1972).
S. A. Chaplygin, “On motion of heavy bodies in an incompressible fluid,” in: Complete Collection of Works, Vol. 1 [in Russian], Izd. Akad. Nauk SSSR, Leningrad (1933), pp. 133–135.
S. A. Chaplygin, Selected Works [in Russian], Nauka, Moscow (1976).
E. Kamke, Differentialgleichungen. I: Gewöhnliche Differentialgleichungen, Akademische Verlagsgesellschaft Geest und Portig K.G., Leipzig (1969).
V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983).
V. V. Kozlov, “Rational integrals of quasihomogeneous dynamical systems,” Prikl. Mat. Mekh., 79, No. 3, 307–316 (2015).
H. Poincaré, On curves defined by differential equations [Russian translation], OGIZ, Moscow-Leningrad (1947).
B. V. Shabat, Introduction to Complex Analysis [in Russian], Nauka, Moscow (1987).
M. V. Shamolin, “Existence and uniqueness of trajectories having infinitely distant points as limit sets for dynamical systems on a plane,” Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., No. 1, 68–71 (1993).
M. V. Shamolin, “Spatial Poincar´e topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997).
M. V. Shamolin, “On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998).
M. V. Shamolin, “A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000).
M. V. Shamolin, Methods of Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007).
M. V. Shamolin, “The problem of a rigid body motion in a resisting medium with the assumption of dependence of the force moment on the angular velocity,” Mat. Model., 24, No. 10, 109–132 (2012).
M. V. Shamolin, “Variety of integrable cases in dynamics of low- and multi-dimensional rigid bodies in nonconservative force fields,” Itogi Nauki i Tekh. Ser. Sovr. Mat. Prilozh. Temat. Obzory, 125, 5–254 (2013).
M. V. Shamolin, “Modeling of the motion of a rigid body in a resisting medium and analogies with vortex paths,” Mat. Model., 27, No. 1, 33–53 (2015).
M. V. Shamolin, “Integrable systems with variable dissipation on the tangent bundle of the sphere,” Probl. Mat. Anal., No. 86, 139–151 (2016).
M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of the two-dimensional and three-dimensional spheres,” Dokl. Ross. Akad. Nauk, 471, No. 5, 547–551 (2016).
M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of two-dimensional manifolds,” Dokl. Ross. Akad. Nauk, 475, No. 5, 519–523 (2017).
M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of three-dimensional manifolds,” Dokl. Ross. Akad. Nauk, 477, No. 2, 168–172 (2017).
M. V. Shamolin, “Integrable dissipative dynamical systems a finite number degrees of freedom,” Probl. Mat. Anal., No. 95, 79–101 (2018).
M. V. Shamolin, “Integrable dissipative systems with two and three degrees of freedom,” Probl. Mat. Anal., No. 94, 91–109 (2018).
M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of multidimensional manifolds,” Dokl. Ross. Akad. Nauk, 482, No. 5, 527–533 (2018).
M. V. Shamolin, “New examples of integrable systems with dissipation on the tangent bundles of four-dimensional manifolds,” Dokl. Ross. Akad. Nauk, 479, No. 3, 270–276 (2018).
M. V. Shamolin, “Examples of integrable systems with dissipation on the tangent bundle of the multidimensional sphere,” Probl. Mat. Anal., No. 90, 107–113 (2018).
V. V. Trofimov and M. V. Shamolin, “Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems,” Fundam. Prikl. Mat., 16, No. 4, 3–229 (2010).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 174, Geometry and Mechanics, 2020.
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Shamolin, M.V. Odd-Order Integrable Dynamical Systems with Dissipation. J Math Sci 272, 672–689 (2023). https://doi.org/10.1007/s10958-023-06464-4
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DOI: https://doi.org/10.1007/s10958-023-06464-4