Adiabatic limits of closed orbits for some Newtonian systems in \(\mathbb{R}^n\) (Q2730804)

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 \(M\).