The compact category and multiple periodic solutions of Hamiltonian systems on symmetric starshaped energy surfaces
We are interested in periodic solutions of the Hamiltonian system (HS) \(\dot x=J\nabla H(x)\) in \(\mathbb{R}^{2N}\) which lie on a fixed regular energy surface \(S=H^{-1}(c)\subset\mathbb{R}^{2N}\). Assume that a compact Lie group \(\Gamma\) acts orthogonally on \(\mathbb{R}^{2N}\) leaving \(S\) invariant and such that each element of \(\Gamma\) acts either as a symplectic or antisymplectic matrix. If \(S\) bounds a compact starshaped neighborhood of \(0\) in \(\mathbb{R}^{2N}\) and satisfies an additional geometric assumption then we prove that (HS) has at least \(N/(1+\dim \Gamma-\dim T\Gamma)\) geometrically distinct \(\Gamma\)-orbits of periodic solutions on \(S\) where \(T\Gamma\) is the maximal torus of \(\Gamma\). We also discuss various examples where the periodic solutions which we obtain are symmetric, like brake orbits or normal mode solutions. These results extend and unify several earlier works on this subject. In the course of the proof we construct a relative equivariant Lusternik-Schnirelmann category and develop a critical point theory for the (strongly indefinite) Hamiltonian action functional. In order to compute the category of the sublevel sets of this functional we prove a new Borsuk-Ulam type theorem for torus actions.
scientific article; zbMATH DE number 4215964
scientific article