Abstract
Consider the planar Newtonian \((2N+1)\)-body problem, \(N\ge 1,\) with \(2N\) bodies of unit mass and one body of mass \(m\). Using the discrete symmetry due to the equal masses and reducing by the rotational symmetry, we show that solutions with the \(2N\) unit mass points at the vertices of two concentric regular \(N\)-gons and \(m\) at the centre at all times form invariant manifold. We study the regular \(2N\)-gon with central mass \(m\) relative equilibria within the dynamics on the invariant manifold described above. As \(m\) varies, we identify the bifurcations, relate our results to previous work and provide the spectral picture of the linearization at the relative equilibria.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Buono, P. L., Laurent-Polz F., Montaldi, J.: Symmetric Hamiltonian bifurcations, In: Mondaldi, J., Ratiu, T. (eds.), Geometric mechanics and symmetry: the peyresq lectures, London Mathematical Society Lecture Note Series vol. 306. Cambridge University Press, Cambridge (2005)
Burgos-Garcia, J., Delgado, J.: On the “blue sky catastrophe” termination on the restricted four-body problem. Celest. Mech. Dyn. Astr. 117, 113–136 (2013)
Dellnitz, M., Melbourne, I., Marsden, J.E.: Generic bifurcations of hamiltonian vector fields with symmetry. Nonlinearity 5, 979–996 (1992)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences. Springer-Verlag, New York (1984)
Guillemin, V., Sternberg, A.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984)
Marsden, J.E.: Lectures in Mechanics. Cambridge University Press, Cambridge (1992)
Meyer, K.R., Schmidt, D.S.: Librations of central configurations and braided Saturn rings. Celest. Mech. Dyn. Astr. 55, 289–303 (1993)
Meyer, K.R., Hall, G.R., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Science. Springer-Verlag, Berlin (2009)
Roberts, G.: Linear Stability in the \(1+N\)-gon relative equilibrium. In: Delgado, J., Lacomba, E., Perez-Chavela E. (eds.), Proceedings of the III International Symposium Hamiltonian Systems and Celestial Mechanics, pp. 303–330. World Scientific, Singapore (2000)
Roberts, G.: Linear stability of the elliptic Lagrangian triangle solutions in the three-body problem. J. Differ. Equ. 182, 191–218 (2002)
Sekiguchi, M.: Bifurcation of central configuration in the \(2N+1\) body problem. Celest. Mech. Dyn. Astr. 90, 355–360 (2004)
Acknowledgments
CS was supported by a NSERC Discovery Grant.
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Dr. Vasile Mioc (1948–2013), researcher in astronomy and celestial mechanics at the Astronomical Institute of the Romanian Academy.
Rights and permissions
About this article
Cite this article
Crînganu, J., Paşca, D. & Stoica, C. A Note on the Relative Equilibria Bifurcations in the \((2N+1)\)-Body Problem. J Dyn Diff Equat 28, 239–251 (2016). https://doi.org/10.1007/s10884-014-9388-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-014-9388-8