Advertisement

Journal of Dynamics and Differential Equations

, Volume 26, Issue 3, pp 529–581 | Cite as

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

  • Jaume Llibre
  • Rafael Ramírez
  • Natalia Sadovskaia
Article

Abstract

This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in \(\mathbb {R}^N\) which have a given set of either \(M\) partial integrals, or \(M<N\) first integral, or \(M<N\) partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of \(N-1\) independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.

Keywords

Algebraic limit circles Polynomial planar differential system Polynomial vector fields Invariant circles Invariant algebraic circles Darboux integrability   16th Hilbert’s problem 

Mathematics Subject Classification

34C07 

Notes

Acknowledgments

The first author is partially supported by a MINECO/FEDER Grant MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2013SGR-568, an ICREA Academia, the Grants FP7-PEOPLE-2012-IRSES 318999 and 316338, FEDER-UNAB10-4E-378, and a CAPES grant number 88881.030454/2013-01 from the program CSF-PVE. The second author was partly supported by the Spanish Ministry of Education through projects TSI2007-65406-C03-01 “AEGIS” and Consolider CSD2007-00004 “ES”.

References

  1. 1.
    Arnold, V.I.: Dynamical Systems III. Springer, Berlin (1996)Google Scholar
  2. 2.
    Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical mechanics. In: Dynamical Systems III. Springer, Berlin (1998)Google Scholar
  3. 3.
    Azcárraga, J.A., Izquierdo, J.M.: \(n\)-ary algebras: a review with applications. J. Phys. A: Math. Theor. 43, 293001 (2010)CrossRefGoogle Scholar
  4. 4.
    Bertrand, M.I.: Sur la possibilité de déduire d’une seule de lois de Kepler le principe de l’attraction. Comptes Rendues 9, 671–674 (1877)Google Scholar
  5. 5.
    Charlier, C.L.: Celestial Mechanics (Die Mechanik Des Himmels) Nauka (1966) (in Russian)Google Scholar
  6. 6.
    Dainelli, U.: Sul movimento per una linea qualunque. Giorn. Mat. 18, 271 (1880)Google Scholar
  7. 7.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Sciencie, NY (1964)Google Scholar
  8. 8.
    Ermakov, V.P.: Determination of the potential function from given partial integrals, Math. Sbornik 15, serie 4 (1881), (in Russian)Google Scholar
  9. 9.
    Erugin, N.P.: Construction of the whole set of differential equations having a given integral curve. Akad. Nauk SSSR. Prikl. Mat. Meh. 16:659–670 (in Russian) (1952)Google Scholar
  10. 10.
    Filippov, V.T.: On the n-Lie algebras of Jacobians. Sibirsk. Mat. Zh. 3, 660–669 (1998)Google Scholar
  11. 11.
    Galiullin, A.S.: Inverse Problems of Dynamics. Mir Publishers (1984)Google Scholar
  12. 12.
    Gantmacher, F.R.: Lektsi po analitisheskoi mechanic, Ed. Nauka, Moscow, 1966 (in Russian)Google Scholar
  13. 13.
    Godbillon, C.: Geometrie Differentielle et Mecanique Analytique. Collection Méthodes Hermann, París (1969)Google Scholar
  14. 14.
    Joukovski, N.E.: Postroenye potencialnaia funksia po zadannie cemiestvo trayectories, Sobranye sochinyeni, T.1, Ed.Gostexizdat, 1948 (in Russian).Google Scholar
  15. 15.
    Kozlov, V.V.: Dynamical Systems X. General Theory of Vortices. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  16. 16.
    Ibañez, R., de León, M., Marreros, J., Padrón, E.: Leibniz algebroid associated with a Nambu–Poisson structure. J. Phys. A: Math. Gen. 32, 8129–8144 (1999)CrossRefMATHGoogle Scholar
  17. 17.
    de Leon, M., de Diego, D.M.: On the geometry of generalized Chaplygin systems. Math. Proc. Camb. Philos. Soc. 132, 1389–1412 (2002)Google Scholar
  18. 18.
    Moser, J.: Various aspects of integrable Hamiltonian systems. In: Helagason, S., Coates, J. (eds.) Dynamical Systems, C.I.M.E. Lectures, Bressanone 1978, Birkhäauser, Boston, 2 edition, 1983, pp. 233–290Google Scholar
  19. 19.
    Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2412 (1973)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Neimark, Ju.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. American Mathematical Society, Rhode Island (1972)Google Scholar
  21. 21.
    Nekhoroshev, N.N.: Variables “action-angle” and their generalizations. Tr. Mosk. Mat. Obshch. 26, 181–198 (1972). (in Russian)MATHGoogle Scholar
  22. 22.
    Newton, I.: Philosophie Naturalis Principia Mathematica, London (1687)Google Scholar
  23. 23.
    Llibre, J., Ramírez, R., Sadovskaia, N.: Integrability of the constrained rigid body. Nonlinear Dyn. 73, 2273–2290 (2013)CrossRefMATHGoogle Scholar
  24. 24.
    Llibre, J., Ramírez, R., Sadovskaia, N.: A new approach in vakonomic mechanics, to appear in Nonlinear DynamicsGoogle Scholar
  25. 25.
    Ramírez, R., Sadovskaia, N.: Inverse problem in Celestial Mechanics. Atti. Sem. Mat. Fis. Univ. Modena e Reggio Emilia LII 47, 47–68 (2004)Google Scholar
  26. 26.
    Ramírez, R., Sadovskaia, N.: Inverse approach into the study of ordinary differential equations, preprint Universitat Rovira i Virgili (2008), pp. 1–49Google Scholar
  27. 27.
    Ramirez, R., Sadovskaia, N.: Differential equations having a complete set of independent first integrals, Arxiv (2011)Google Scholar
  28. 28.
    Sadovskaia, N.: Inverse problem in theory of ordinary differential equations, Thesis Ph. D., Univ. Politécnica de Cataluña, 2002 (in Spanish)Google Scholar
  29. 29.
    Sadovskaia, N., Ramírez, R.: Differential equations of first order with given set of partial integrals, Technical Report MA II-IR-99-00015 Universitat Politecnica de Catalunya, (1999) (in Spanish)Google Scholar
  30. 30.
    Sundermeyer, K.: Constrained Dynamics. Lecture Notes in Physics, vol. 169. Spriger, NY (1982)Google Scholar
  31. 31.
    Suslov, G.K.: On a particular variant of d’Alembert principle. Math. Sb. 22, 687–691 (1901). (in Russian)Google Scholar
  32. 32.
    Suslov, G.K.: Opredilienyie silovoi funkcyi po zadanniem shasnim integralam. Ph.D, Kiev (in Russian) (1890)Google Scholar
  33. 33.
    Suslov, G.K: Theoretical Mechanics. M.-L.: Gostechizdat (1946)Google Scholar
  34. 34.
    Synge, J.L.: On the geometry of dynamics. Philos. Trans. R. Soc. Lond. A 226, 31–106 (1926)CrossRefMATHGoogle Scholar
  35. 35.
    Szebehely, V.: Open problems on the eve of the next millenium. Celest. Mech. 65, 205–211 (1996)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160, 295–315 (1994)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Whittaker, E.T.: A Treatise on the Analytic Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1959)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jaume Llibre
    • 1
  • Rafael Ramírez
    • 2
  • Natalia Sadovskaia
    • 3
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain
  3. 3.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations