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Adiabatic limits of closed orbits for some Newtonian systems in \(\mathbb{R}^n\). (English) Zbl 0988.37076

Long-period orbits in an \(n\)-dimensional potential \(V\) possessing an \((n-1)\)-dimensional non-degenerate manifold \(M\) of equilibrium points are examined. It is known that there exist monoparametric families of long-periodic orbits with respect to the period \(T\) and the limiting curve of such a family as \(T \rightarrow \infty\) is a geodesic on \(M\). If \(M\) consists of unstable equilibria, it is shown that if \(x_0\) is a non-degenerate closed geodesic on \(M\), a family of periodic solutions exists, whose limit as \(T \rightarrow \infty\) is \(x_0\). Conversely, for every unbounded sequence \(T_k\), there exist periodic solutions with periods \(T_k\) which converge to a non-trivial closed geodesic on \(M\). In the case of stable equilibria there are problems due to resonances. In this case it is shown that, under some stronger assumptions, there exists a sequence \(T_k\), such that periodic solutions with periods \(T_k\) exist, which converge to a non-trivial closed geodesic on \(M\).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
70H03 Lagrange’s equations
34C25 Periodic solutions to ordinary differential equations
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