Astronomy and Astrophysics – Astronomy
Scientific paper
Mar 2005
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2005cemda..91..269m&link_type=abstract
Celestial Mechanics and Dynamical Astronomy, Volume 91, Issue 3-4, pp. 269-285
Astronomy and Astrophysics
Astronomy
Schwarzschild-Type Problems, Nonlinear Particle Dynamics, Symmetries, Periodic Orbits, Variational Methods
Scientific paper
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).
Anisiu Mira-Cristiana
Barbosu Michael
Mioc Vasile
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