Skip to main content
SpringerLink
Log in

On the Correspondence between the Variational Principles in the Eulerian and Lagrangian Descriptions

  • Research Articles
  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

The relationship between the variational principles for equations of continuum mechanics in Eulerian and Lagrangian descriptions is considered. It is shown that, for a system of differential equations in Eulerian variables, the corresponding Lagrangian description is related to introducing nonlocal variables. The connection between the descriptions is obtained in terms of differential coverings. The relation between the variational principles of a system of equations and its symplectic structures is discussed. It is shown that, if a system of equations in Lagrangian variables can be derived from a variational principle, then there is no corresponding variational principle in the Eulerian variables.

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. A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor’kova, I.S. Krasil’shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, and A.M. Vinogradov, “Symmetries and Conservation Laws for Differential Equations of Mathematical Physics”, Amer. Math. Soc., 182 (1999).

    MathSciNet  MATH  Google Scholar 

  2. A.V. Aksenov and K.P. Druzhkov, “Conservation Laws of the Equation of One-Dimensional Shallow Water Over Uneven Bottom in Lagrange’s Variables”, Internat. J. Non-Linear Mech., 119 (2020), 103348.

    Article  ADS  Google Scholar 

  3. V.L. Berdichevsky, Variational Principles of Continuum Mechanics. I. Fundamentals, Springer-Verlag, Berlin, Heidelberg, Berlin, 2009.

    Book  Google Scholar 

  4. F. Bampi and A. Morro, “The Connection Between Variational Principles in Eulerian and Lagrangian Descriptions”, J. Math. Phys., 25 (1984).

    Article  MathSciNet  Google Scholar 

  5. I. Khavkine, “Presymplectic Current and the Inverse Problem of the Calculus of Variations”, J. Math. Phys., 54 (2013).

    Article  MathSciNet  Google Scholar 

  6. K.P. Druzhkov, “Extendable Symplectic Structures and the Inverse Problem of the Calculus of Variations for Systems of Equations Written in Generalized Kovalevskaya Form”, J. Geom. Phys., 161 (2021).

    Article  MathSciNet  Google Scholar 

  7. V.A. Dorodnitsyn, E.I. Kaptsov, and S.V. Meleshko, “Symmetries, Conservation Laws, Invariant Solutions and Difference Schemes of the One-Dimensional Green-Naghdi Equations”, J. Nonlinear Math. Phys., 28 (2021), 90–107.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Mechanics and Mathematics Department, Leninskie Gory 1, 119234, Moscow, Russia

    A. V. Aksenov & K. P. Druzhkov

  2. National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409, Moscow, Russia

    A. V. Aksenov

  3. Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, 141700, Moscow, Russia

    K. P. Druzhkov

Authors
  1. A. V. Aksenov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. P. Druzhkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. V. Aksenov or K. P. Druzhkov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aksenov, A.V., Druzhkov, K.P. On the Correspondence between the Variational Principles in the Eulerian and Lagrangian Descriptions. Russ. J. Math. Phys. 28, 411–415 (2021). https://doi.org/10.1134/S1061920821040014

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920821040014

Search

Navigation