ABSTRACT

A superquadratic Hamiltonian is a function H (z ) which behaves as \z\e, for some 6 > 2, when z lies outside from a large ball in R 2iV. Although su­ perquadratic Hamiltonians may not seem physically interesting (in particu­ lar, it is the super quadratic growth in the p variables which seems unphysi­ cal) they constitute important mathematical examples. Their main feature is that the corresponding action functional satisfies a useful compactness condition, the Palais-Smale condition. This fact enabled Rabinowitz to prove the first variational results about existence of periodic solutions of such systems [Rab78a]. Moreover, many other problems, such as finding closed characteristics on energy surfaces or computing symplectic capaci­ ties, can be handled by introducing suitable superquadratic Hamiltonians.