Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem |
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Authors: | Vasile Mioc Mira-Cristiana Anisiu Michael Barbosu |
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Institution: | (1) Astronomical Institute of the Romanian Academy, Str. Cutitul de Argint 5, RO-040557 Bucharest, Romania;(2) T. Popoviciu Institute of Numerical Analysis of the Romanian Academy, P.O. Box 68, 400110 Cluj-Napoca, Romania;(3) SUNY Brockport, Department of Mathematics, Brockport, NY, 14420, U.S.A. |
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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
,
, 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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Keywords: | Schwarzschild-type problems nonlinear particle dynamics symmetries periodic orbits variational methods |
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