Skip to main content
SpringerLink
Log in

A brief note on the analytical solution of Meshchersky’s equation within the inverse problem of Lagrangian mechanics

  • Note
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Meshchersky’s equation is a basic differential equation in the mechanics of variable-mass particles. This note particularly considers the case in which a one-dimensional and position-dependent mass particle is under the action of a potential force. The absolute velocity of mass ejection (or accretion) is supposed to be a linear function of the particle velocity. Within the formulation of the inverse problem of Lagrangian mechanics, an analytical solution of Meshchersky’s equation is here derived. The solution method follows from applying the concept of constant of motion of an extremum problem, which is a fundamental ground in the theory of invariant variational principles.

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

References

  1. Irschik H., Holl H.J.: Mechanics of variable-mass systems—part 1: balance of mass and linear momentum. Appl. Mech. Rev. 57, 145–160 (2004)

    Article  Google Scholar 

  2. Poisson, S.D.: Sur le mouvement d’un système de corps, en supposant les masses variables. Bull. Sci. Soc. Philomat. Paris 60–62 (1819)

  3. Cayley A.: On a class of dynamical problems. Proc. R. Soc. Lond. 8, 506–511 (1857)

    Article  Google Scholar 

  4. Mikhailov G.K.: On the history of variable-mass system dynamics. Mech. Solids 10, 32–40 (1975)

    Google Scholar 

  5. Leach P.G.L.: Harmonic oscillator with variable mass. J. Phys. A Math. Gen. 16, 3261–3269 (1983)

    Article  MathSciNet  Google Scholar 

  6. Cveticanin L.: Approximate solution of a time-dependent differential equation. Meccanica 30, 665–671 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Irschik, H., Belyaev, A.K. (eds.): Dynamics of mechanical systems with variable mass. In: Series: CISM International Centre for Mechanical Sciences, vol. 557, 266 p. Springer, Berlin (2014)

  8. Casetta L., Pesce C.P.: The inverse problem of Lagrangian mechanics for Meshchersky’s equation. Acta Mech. 225, 1607–1623 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Casetta, L.: The inverse problem of Lagrangian mechanics for a non-material volume. Acta Mech. 226, 1–15 (2015)

  10. Helmholtz H.: Über die physikalische Bedeutung des Princips der kleinsten Wirkung. J. Reine Angew. Math. 100, 137–166 (1887)

    Google Scholar 

  11. Havas P.: The range of application of the Lagrange formalism—I. Suppl. Nuovo Cim. V(X), 363–388 (1957)

    Article  MathSciNet  Google Scholar 

  12. Santilli R.M.: Foundations of Theoretical Mechanics I. The Inverse Problem in Newtonian Mechanics. Springer, New York (1978)

    Book  MATH  Google Scholar 

  13. Pesce C.P.: The application of Lagrange equations to mechanical systems with mass explicitly dependent on position. J. Appl. Mech. 70, 751–756 (2003)

    Article  MATH  Google Scholar 

  14. Goldstein H., Poole C.P., Safko J.L.: Classical Mechanics. Addison-Wesley, San Francisco (2002)

    Google Scholar 

  15. Whittaker E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1988)

    Book  MATH  Google Scholar 

  16. Logan J.D.: Invariant Variational Principles. Academic Press, New York (1977)

    Google Scholar 

  17. Neuenschwander D.E.: Emmy Noether’s Wonderful Theorem. The Johns Hopkins University Press, Baltimore (2011)

    MATH  Google Scholar 

  18. Sarlet W., Cantrijn F.: Generalizations of Noether’s theorem in classical mechanics. SIAM Rev. 23, 467–494 (1981)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Offshore Mechanics Laboratory, Department of Mechanical Engineering, Escola Politécnica, University of São Paulo, São Paulo, Brazil

    Leonardo Casetta & Celso P. Pesce

Authors
  1. Leonardo Casetta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Celso P. Pesce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonardo Casetta.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Casetta, L., Pesce, C.P. A brief note on the analytical solution of Meshchersky’s equation within the inverse problem of Lagrangian mechanics. Acta Mech 226, 2435–2439 (2015). https://doi.org/10.1007/s00707-015-1315-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-015-1315-5

Keywords

Search

Navigation