Morse theory and asymptotic linear Hamiltonian system
We try to quote from the Introduction in order to give an idea on the paper avoiding technical details. ``We consider the existence of \(2\pi\)- periodic solutions of Hamiltonian systems \(-J\dot z=H'(z,t)\) \((*)\quad | H'(s,t)-h_{\infty}s| /| s| \to 0,\) as \(| s| \to \infty\) \(| H'(s,t)-h_ 0s| /| s| \to 0,\) as \(| s| \to 0\) \((h_{\infty},h_ 0\) are constant symmetric matrices and \(H'\) denotes the gradient of H with respect to the first 2n variables). Given a constant 2n\(\times 2n\) matrix h we define an index \(i^-(h)\). measuring the difference between the ``sizes'' of the negative spaces of the operators -Jd/dt-h and -Jd/dt. We define \(i^ 0(h)\) which is just the dimension of the null space of -Jd/dt-h. Let \(H: R^{2n}\times R\to R\) be \(C^ 1,2\pi\)-periodic in t and satisfying (*). If \(i^ 0(h_ 0)=i^ 0(h_{\infty})=0\) and \(i^-(h_ 0)\neq i^-(h_{\infty})\) then there exists at least one nontrivial periodic solution''. Morse theory and Galerkin method are combined.
scientific article; zbMATH DE number 4099723
scientific article