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Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem

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Abstract

Studying the two-body problem associated to an anisotropic Schwarzschild-type field, Mioc et al. (2003) did not succeed in proving the existence or non-existence of periodic orbits. Here we answer this question in the affirmative. To do this, we start from two basic facts: (1) the potential generates a strong force in Gordon’s sense; (2) the vector field of the problem exhibits the symmetries S i , \(i =\overline {1, 7} \), which form, along with the identity, an Abelian group of order 8 with three generators of order 2. Resorting to S 2 and S 3, in connection with variational methods (particularly the classical lower-semicontinuity method), we prove the existence of infinitely many S 2- or S 3-symmetric periodic solutions. The symmetries S 2 and S 3 constitute an indicator of the robustness of the classical isotropic Schwarzschild-type system to perturbations (as the anisotropy may be considered).

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Authors and Affiliations

  1. Astronomical Institute of the Romanian Academy, Str. Cutitul de Argint 5, RO-040557, Bucharest, Romania

    Vasile Mioc

  2. T. Popoviciu Institute of Numerical Analysis of the Romanian Academy, P.O. Box 68, 400110, Cluj-Napoca, Romania

    Mira-Cristiana Anisiu

  3. SUNY Brockport, Department of Mathematics, Brockport, NY, 14420, U.S.A.

    Michael Barbosu

Authors
  1. Vasile Mioc
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  2. Mira-Cristiana Anisiu
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  3. Michael Barbosu
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Correspondence to Vasile Mioc.

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Mioc, V., Anisiu, MC. & Barbosu, M. Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem. Celestial Mech Dyn Astr 91, 269–285 (2005). https://doi.org/10.1007/s10569-004-1721-9

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  • DOI: https://doi.org/10.1007/s10569-004-1721-9

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