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

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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.

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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”.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain

    Jaume Llibre

  2. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avinguda dels Països Catalans 26, 43007, Tarragona, Catalonia, Spain

    Rafael Ramírez

  3. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Pau Gargallo 5, 08028, Barcelona, Catalonia, Spain

    Natalia Sadovskaia

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  1. Jaume Llibre
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  2. Rafael Ramírez
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  3. Natalia Sadovskaia
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Correspondence to Jaume Llibre.

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Llibre, J., Ramírez, R. & Sadovskaia, N. Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics. J Dyn Diff Equat 26, 529–581 (2014). https://doi.org/10.1007/s10884-014-9390-1

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  • DOI: https://doi.org/10.1007/s10884-014-9390-1

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