A class of periodic solutions of the \(N\)-body problem. (English) Zbl 0684.70006

Coti Zelati, Vittorio

Celest. Mech. Dyn. Astron. 46, No. 2, 177-186 (1989).

Summary: We prove existence and multiplicity of \(T\)-periodic solutions (for any given \(T\)) for the \(N\)-body problem in \(\mathbb R^m\) (any \(m\geq 2)\) where one of the bodies has mass equal to 1 and the others have masses \(\varepsilon \alpha_ 2,\ldots,\epsilon \alpha_ N\), \(\varepsilon\) small. We find solutions such that the body of mass 1 moves close to \(x=0\) while the body of mass \(\varepsilon \alpha_ i\) moves close to one of the circular solutions of the two body problem of period \(T/k_ i\), where \(k_ i\) is any odd number. No relation has to be satisfied by \(k_ 2,\ldots,k_ N\).

MSC:

70F10	\(n\)-body problems
37G99	Local and nonlocal bifurcation theory for dynamical systems
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

small parameter method; multiplicity of T-periodic solutions; N-body problem; circular solutions

References:

[1]	Ambrosetti, A.; Zelati, V. Coti; Ekeland, I., Symmetry breaking in Hamiltonian systems, J. Dif: Eq., 67, 165-184, (1987) · Zbl 0606.58043
[2]	A. Ambrosetti and V. Coti Zelati: ‘Perturbation of Hamiltonian systems with Keplerian potentials’, Mathematische Zeitschrift, in print. · Zbl 0653.34032
[3]	A. Ambrosetti and V. Coti Zelati: 1988, ‘Oscillations non-linéaires pour des problémes de la Mécanique Céleste’, C. R. Acad. Sci. Paris307, Série I, 569-571. · Zbl 0653.70013
[4]	Gordon, W., A minimizing property of Keplerian orbits, Am. J. Math., 99, 961-971, (1977) · Zbl 0378.58006
[5]	Meyer, K. R., Periodic solutions of the \(N\)-body problem, J. Diff. Eq., 39, 2-38, (1981) · Zbl 0431.70021