Abstract:
This paper discusses conditions for the existence of polynomial (in velocities)
first integrals of the equations of motion of mechanical systems in a nonpotential force field
(circulatory systems). These integrals are assumed to be single-valued smooth functions on
the phase space of the system (on the space of the tangent bundle of a smooth configuration
manifold). It is shown that, if the genus of the closed configuration manifold of such a system
with two degrees of freedom is greater than unity, then the equations of motion admit no
nonconstant single-valued polynomial integrals. Examples are given of circulatory systems with
configuration space in the form of a sphere and a torus which have nontrivial polynomial laws
of conservation. Some unsolved problems involved in these phenomena are discussed.
Keywords:
circulatory system, polynomial integral, genus of surface.
\Bibitem{Koz22}
\by Valery V. Kozlov
\paper On the Integrability of Circulatory Systems
\jour Regul. Chaotic Dyn.
\yr 2022
\vol 27
\issue 1
\pages 11--17
\mathnet{http://mi.mathnet.ru/rcd1149}
\crossref{https://doi.org/10.1134/S1560354722010038}
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This publication is cited in the following 6 articles:
Ivan Yu. Polekhin, “On the dynamics and integrability of the Ziegler pendulum”, Nonlinear Dyn, 112:9 (2024), 6847
Elizaveta Artemova, Evgeny Vetchanin, “The motion of a circular foil in the field of a fixed point singularity: Integrability and asymptotic behavior”, Physics of Fluids, 36:2 (2024)
Stefano Disca, Vincenzo Coscia, “Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum”, Meccanica, 59:7 (2024), 1139
Marwa Banna, Tobias Mai, “Berry-Esseen bounds for the multivariate ℬ-free CLT and operator-valued matrices”, Trans. Amer. Math. Soc., 376:6 (2023), 3761
Davide Bigoni, Francesco Dal Corso, Oleg N. Kirillov, Diego Misseroni, Giovanni Noselli, Andrea Piccolroaz, “Flutter instability in solids and structures, with a view on biomechanics and metamaterials”, Proc. R. Soc. A., 479:2279 (2023)
V. V. Kozlov, “On the integrability of the equations of dynamics in a non-potential force field”, Russian Math. Surveys, 77:6 (2022), 1087–1106
Citation in format AMSBIB
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