Several problems in the field of differential equations will be under investigation during the term of this project. They concern systems of differential equations from the physical sciences, namely Lagrangian equations representing forced motion of a mechanical system and the more general Hamiltonian systems. A system of forced pendulums would give rise to the first class of equations, whereas the motion of bodies under gravitational forces often uses the Hamiltonian structure. The Lagrangian equations involved in this study have periodic coefficients and forcing functions. It is natural to inquire whether of not there will exist periodic solutions under such assumptions. As with many equations modeling phenomena from the physical world, the Lagrangians result from a principle of energy or action minimization. One seeks to find critical points of a certain integral with admissible functions taken from a Hilbert space. Depending on the growth of the potential, there may be many distinct periodic solutions, whereas evidence suggests that for periodic potentials, only finitely many periodic solutions may exist. A measure of the number of such solutions in terms the structure of the potential will be sought. Work on Hamiltonian systems will follow a similar vein concentrating on the case of periodic Hamiltonians. Solutions of these systems are known as subharmonics. There are known to be periodic subharmonics under very general conditions. The most important subharmonics are those with minimal periods. This research will look for a description of solutions with minimal periods in cases where additional symmetry assumptions are made on the Hamiltonian. Solutions of Hamiltonian systems must lie on surfaces of constant energy. A related goal of this work will be to determine which surfaces of the Hamiltonian carry a periodic orbit.
