A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions

Abstract

In this paper, we study the multiplicity problem for Euler–Lagrange orbits that satisfy the conormal boundary conditions for a suitable class of reversible Lagrangian functions on compact manifolds. Such a class contains, e.g. the energy function of reversible Finsler metrics that satisfy a convexity condition on the boundary.

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Notes

  1. 1.

    Two curves \(\gamma _1,\gamma _2: [0,1] \rightarrow {\overline{\Omega }}\) are considered distinct if \(\gamma _1([0,1]) \ne \gamma _2([0,1])\).

  2. 2.

    There is a standard construction of metrics for which a given closed embedded submanifold of a differentiable manifold is totally geodesic. Such metrics are constructed in a tubular neighbourhood first, using a normal bundle construction, and then extended using a partition of unity argument.

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Acknowledgements

I wish to thank the referee for his/her useful suggestions that improved the paper.

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Correspondence to Dario Corona.

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Corona, D. A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions. J. Fixed Point Theory Appl. 22, 60 (2020). https://doi.org/10.1007/s11784-020-00795-4

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Keywords

  • Lagrange’s equations
  • variational methods
  • holonomic systems

Mathematics Subject Classification

  • 70H03
  • 70G75
  • 70F20