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International Conference on Dynamics, Games and Science

DGS 2014: Modeling, Dynamics, Optimization and Bioeconomics II pp 223–265Cite as

Direct and Inverse Variational Problems on Time Scales: A Survey

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 195))

Abstract

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler–Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler–Lagrange equation (Helmholtz’s problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field.

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Acknowledgements

This work was partially supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA), and The Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013. The authors are grateful to two anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193, Aveiro, Portugal

    Monika Dryl & Delfim F. M. Torres

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  1. Monika Dryl
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  2. Delfim F. M. Torres
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Correspondence to Delfim F. M. Torres .

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  1. Department of Mathematics and LIAAD-INESC TEC, University of Porto, Porto, Portugal

    Alberto A. Pinto

  2. Department of Agricultural and Resource Economics, University of California, Berkeley, California, USA

    David Zilberman

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Dryl, M., Torres, D.F.M. (2017). Direct and Inverse Variational Problems on Time Scales: A Survey. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics II. DGS 2014. Springer Proceedings in Mathematics & Statistics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-55236-1_12

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