The study of periodic orbits in Hamiltonian systems is of great mathematical and physical interest and has been the subject of much effort over the past century. In particular, an important basic problem is the determination of how many periodic orbits a dynamical system has, or, equivalently, the "probability" that a given initial condition leads to a periodic orbit. In many classical systems, such as billiards, it is conjectured that the probability that an orbit is periodic is zero. Surprisingly, this classical mechanics problem is also related to the asymptotic distribution of eigenvalues in the corresponding Helmholtz problem. The structure of the set of periodic orbits also plays important role in the study of quantum chaos and in physics of microcavities. For example, some special sets of periodic orbits support quasimodes, which have been used in microlasers.
We expect the results of this research to help elucidate the general understanding of the structure of periodic orbits in Hamiltonian systems. The proposed research will help physicists working on spectral problems and on the physics of microlasers, which have important applications in materials science and medicine. Graduate students will participate in the research project and broaden their education through interactions with industrial researchers.
