Indices and Stability of the Lagrangian System on Riemannian Manifold

Abstract

In this paper, let m ≥ 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincaré map of the Lagrangian system at critical point x

$${d \over {dt}}{L_q}\left( {t,x,\dot x} \right) - {L_p}\left( {t,x,\dot x} \right) = 0$$

is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.

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Acknowledgements

I would like to sincerely thank Professor Hui Liu for his valuable discussions and careful reading of the manuscript.

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Correspondence to Gao Sheng Zhu.

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Supported by NSFC (Grant Nos. 11871356, 11871368)

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Zhu, G.S. Indices and Stability of the Lagrangian System on Riemannian Manifold. Acta. Math. Sin.-English Ser. (2020). https://doi.org/10.1007/s10114-020-9311-7

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Keywords

  • Nonorientable
  • spectral flow
  • Morse index
  • Maslov-type indices
  • Lagrangian system
  • Hamiltonian system

MR(2010) Subject Classification

  • 58E05
  • 37J45
  • 34C25