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


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.


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

Mathematics Subject Classification




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”.


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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

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