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 Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 6, Pages 45–80 (Mi izv7796)

Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom

A. V. Dymov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider the problem of potential interaction between a finite-dimensional linear Lagrangian system and an infinite-dimensional one (a system of linear oscillators and a thermostat). We study the final dynamics of the system. Under natural assumptions, this dynamics turns out to be very simple and admits an explicit description because the thermostat produces an effective dissipation despite the energy conservation and the Lagrangian nature of the system. We use the methods of [1], where the final dynamics of the finite-dimensional subsystem is studied in the case when it has one degree of freedom and a linear potential or (under additional assumptions) polynomial potential. We consider the case of finite-dimensional subsystems with arbitrarily many degrees of freedom and a linear potential and study the final dynamics of the system of oscillators and the thermostat. The necessary assertions from [1] are given with proofs adapted to the present situation.

Keywords: Lagrangian systems, Hamiltonian systems, systems with infinitely many degrees of freedom, final dynamics.

DOI: https://doi.org/10.4213/im7796

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English version:
Izvestiya: Mathematics, 2012, 76:6, 1116–1149

Bibliographic databases:

UDC: 517.937+517.938
MSC: 37K45, 37L15, 42A38, 46F12, 70H14
Revised: 29.12.2011

Citation: A. V. Dymov, “Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom”, Izv. RAN. Ser. Mat., 76:6 (2012), 45–80; Izv. Math., 76:6 (2012), 1116–1149

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv7796
• https://doi.org/10.4213/im7796
• http://mi.mathnet.ru/eng/izv/v76/i6/p45

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. Dymov, “Nonequilibrium statistical mechanics of Hamiltonian rotators with alternated spins”, J. Stat. Phys., 158:4 (2015), 968–1006
2. A. V. Dymov, “Nonequilibrium statistical mechanics of a solid immersed in a continuum”, Proc. Steklov Inst. Math., 295 (2016), 95–128
3. Dymov A., “Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed System of Oscillators”, Ann. Henri Poincare, 17:7 (2016), 1825–1882
4. S. M. Saulin, “Dissipation effects in infinite-dimensional Hamiltonian systems.”, Theoret. and Math. Phys., 191:1 (2017), 537–557
5. Dymov A.V., “Asymptotic Behavior of a Network of Oscillators Coupled to Thermostats of Finite Energy”, Russ. J. Math. Phys., 25:2 (2018), 183–199
6. A. I. Komech, E. A. Kopylova, “Attraktory nelineinykh gamiltonovykh uravnenii v chastnykh proizvodnykh”, UMN, 75:1(451) (2020), 3–94
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