Closed characteristics on non-compact hypersurfaces in \(\mathbb {R}^{2n}\)
The authors consider mechanical Hamiltonians \(H(p,q)=1/2 |p|^2 +V(q)\) on the standard symplectic vector space \(({\mathbb R}^{2n}, \omega).\) Assuming that \(0\) is a regular value of \(H\) and the level hypersurface \(M=H^{-1}(0)\) asymptotically regular they prove that \(M\) contains at least one closed characteristic of the Hamiltonian vectorfield associated to \(H\) provided that the homology of \(M\) is nontrivial on dimensions greater or equal to \(n\). Roughly speaking, the asymptotic regularity requires that the growth of the Hessian of the potential function \(V\) is dominated by that of the gradient of \(V\) at infinity. This condition is needed in order to be able to deal with non-compact hypersurfaces. The proof of the above result, as many others of this type, relies on a linking principle. However due to the lack of compactness of \(M\) the technical details are considerably more involved than in the compact case.
scientific article; zbMATH DE number 5503660
scientific article