Models for the motion of a finite number of particlies where no frictional forces are present are usually given by means of differential equations called Hamiltonian systems. During the past decade there has been intensive research focusing on the existence of periodic solutions of such systems. The work sponsored by this award is in part an outgrowth of a more general development of variational methods, expecially minimax and Morse theoretic methods, and their application to nonlinear differential equations. It illustrates the effectiveness which can be achieved when the power of abstract mathematical research is driven by external stimuli. One focus of this work concerns the Hamiltonian itself, a function of two (vector) variables which contains the position and velocity information about the particles. Traditionally one assumes that the Hamiltonians are continuously differentiable. Many interesting problems, however, involve singular Hamitonians. The classical n-body problem of celestial mechanics is an example. Using minimax arguments, one can show that there are infinitely many periodic solutions with distinct periods. The present work will seek to clarify whether or not multiple periodic solutions can exist with the same period. In addition, with the advent of singular Hamiltonians, the possibility arises for collision orbits. Work will be done in determining conditions where one can distinguish between regular and collision orbits. Efforts will also be made to find periodic solutions with prescribed energy. Branching away from periodic solutions, research will begin on investigations into the next simplest class, those orbits which tend to equilibrium solutions as times increases indefinitely. These are known as connecting orbits, for they join period ones. Some results are available for so-called superquadratic Hamiltonian systems, but the area is still in its infancy.
