American Institute of Mathematical Sciences

Advanced
December  2010, 28(4): 1693-1712. doi: 10.3934/dcds.2010.28.1693

Oscillator and thermostat

Dmitry Treschev 1,

1. 

Steklov Mathematical Institute, Gubkina str., Moscow, 119991

Received  October 2009 Revised  February 2010 Published  June 2010

We study the problem of a potential interaction of a finite- dimensional Lagrangian system (an oscillator) with a linear infinite-dimensional one (a thermostat). In spite of the energy preservation and the Lagrangian (Hamiltonian) nature of the total system, under some natural assumptions the final dynamics of the finite-dimensional component turns out to be simple while the thermostat produces an effective dissipation.
Keywords: Hamiltonian systems, final dynamics., Lagrangian systems.
Mathematics Subject Classification: Primary: 37K; Secondary: 70.
Citation: Dmitry Treschev. Oscillator and thermostat. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1693-1712. doi: 10.3934/dcds.2010.28.1693
[1]

B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217
[2]

Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
[3]

Yangyou Pan, Yuzhen Bai, Xiang Zhang. Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1761-1774. doi: 10.3934/dcdss.2019116
[4]

Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201
[5]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
[6]

Anouar Bahrouni, Marek Izydorek, Joanna Janczewska. Subharmonic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1841-1850. doi: 10.3934/dcdss.2019121
[7]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213
[8]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204
[9]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130
[10]

Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49
[11]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of discrete mechanical systems by stages. Journal of Geometric Mechanics, 2016, 8 (1) : 35-70. doi: 10.3934/jgm.2016.8.35
[12]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69
[13]

E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1
[14]

Francesca Alessio, Vittorio Coti Zelati, Piero Montecchiari. Chaotic behavior of rapidly oscillating Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 687-707. doi: 10.3934/dcds.2004.10.687
[15]

Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1
[16]

Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[17]

Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069
[18]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[19]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
[20]

K. Tintarev. Critical values and minimal periods for autonomous Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 389-400. doi: 10.3934/dcds.1995.1.389

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences