Skip to main content
SpringerLink
Log in

On Dissipative Phenomena of the Interaction of Hamiltonian Systems

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

The dynamics is under study of a composite Hamiltonian system that is the union of a finite-dimensional nonlinear system and an infinite-dimensional linear system with quadratic interaction Hamiltonian. The dynamics of the finite-dimensional subsystem is determined by a nonlinear integro-differential equation with a relaxation kernel. We prove existence and uniqueness theorems and find a priori estimates for a solution. Under some assumptions on the form of interaction, the solution to the finite-dimensional subsystem converges to one of the critical points of the effective Hamiltonian.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnol′d V. I., Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  2. Komech A. I., “Attractors of nonlinear Hamilton one-dimensional wave equations,” Uspekhi Mat. Nauk, 55, No. 1, 45–98 (2000).

    Google Scholar 

  3. Dinariev O. Yu., “The relation between the mechanics of dissipative finite-dimensional systems with heredity and the mechanics of infinite-dimensional Hamiltonian systems,” Prikl. Mat. i Mekh., 63, No. 2, 245–257 (1999).

    Google Scholar 

  4. Maurin K., Methods of Hilbert Space [Russian translation], Mir, Moscow (1965).

    Google Scholar 

  5. Dunford N. and Schwartz J. T., Linear Operators. Vol. 2: Spectral Theory [Russian translation], Mir, Moscow (1966).

    Google Scholar 

  6. Dinariev O. Yu., “On the dynamics of a nonlinear multidimensional oscillator with memory,” Sibirsk. Mat. Zh., 41, No. 3, 591–601 (2000).

    Google Scholar 

  7. Hörmander L., The Analysis of Linear Partial Differential Operators. Vol. 1: Distribution Theory and Fourier Analysis [Russian translation], Mir, Moscow (1986).

    Google Scholar 

  8. Hale J. K., Theory of Functional-Differential Equations [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  9. Hartman P., Ordinary Di_erential Equations [Russian translation], Mir, Moscow (1970).

    Google Scholar 

  10. Day W. A., The Thermodynamics of Materials with Memory [Russian translation], Mir, Moscow (1974).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shmidt United Institute of Earth Physics, Moscow

    O. Yu. Dinariev

Authors
  1. O. Yu. Dinariev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dinariev, O.Y. On Dissipative Phenomena of the Interaction of Hamiltonian Systems. Siberian Mathematical Journal 44, 61–72 (2003). https://doi.org/10.1023/A:1022012304018

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022012304018

Search

Navigation