Periodic solutions for Hamiltonian systems under strong constraining forces (Q1867237)

Hamiltonian systems in a Euclidian space, whose motions are constrained to an \(m\)-dimensional submanifold \(M\) are considered. The motion in the ambient space \(\mathbb{R}^{m+l}\), under the action of a strong restoring potential which vanishes on the manifold \(M\) and depends on a parameter \(\varepsilon\), is studied. For a nondegenerate periodic motion \(p_0 \in M\), i.e.\ a periodic motion with only one pair of zero characteristic exponents, it is shown that a sequence of parameter values \(\varepsilon_k \rightarrow 0\) exists, such that, for every \(\varepsilon_k\), a unique nondegenerate periodic solution in the ambient space exists, which is \(O(\varepsilon_k^2)\)-close in the coordinates and \(O(\varepsilon_k)\)-close in the momenta to \(p_0\).