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Trace Formula for Linear Hamiltonian Systems with its Applications to Elliptic Lagrangian Solutions

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Abstract

In the present paper, we build up trace formulas for both the linear Hamiltonian systems and Sturm–Liouville systems. The formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the Hessian of the action functional. A natural application is to study the non-degeneracy of linear Hamiltonian systems. Precisely, by the trace formula, we can give an estimation for the upper bound such that the non-degeneracy preserves. Moreover, we could estimate the relative Morse index by the trace formula. Consequently, a series of new stability criteria for the symmetric periodic orbits is given. As a concrete application, the trace formula is used to study the linear stability of elliptic Lagrangian solutions of the classical planar three-body problem, which depends on the mass parameter \({\beta \in [0,9]}\) and the eccentricity \({e \in [0,1)}\) . Based on the trace formula, we estimate the stable region and hyperbolic region of the elliptic Lagrangian solutions.

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Authors and Affiliations

  1. Department of Mathematics, Shandong University Jinan, Shandong, 250100, The People’s Republic of China

    Xijun Hu, Yuwei Ou & Penghui Wang

Authors
  1. Xijun Hu
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  2. Yuwei Ou
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  3. Penghui Wang
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Correspondence to Xijun Hu.

Additional information

Communicated by P. Rabinowitz

X. Hu was partially supported by NSFC, NCET. Y. Ou was partially supported by NSFC (No.11131004). P. Wang was partially supported by NSFC (No.11101240, No.11471189) and Excellent Young Scientist Foundation of Shandong Province.

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Hu, X., Ou, Y. & Wang, P. Trace Formula for Linear Hamiltonian Systems with its Applications to Elliptic Lagrangian Solutions. Arch Rational Mech Anal 216, 313–357 (2015). https://doi.org/10.1007/s00205-014-0810-5

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