The Dynamics of Hamiltonian Lattices With Application to Hollomon Oscillators

Abstract

Many problems in theoretical physics are expressed in the form of Hamiltonian systems. Among these the first to be extensively studied were low-dimensional, possessing as few as two (or three) degrees of freedom. In the last decades, however, great attention has been devoted to Hamiltonian systems of high dimensionality. The most famous among them are the ones that deal with the dynamics and statistics of a large number N of mass particles connected with nearest neighbor interactions. At low energies E, these typically execute quasiperiodic motions near some fundamental stable periodic orbits (SPOs) which represent nonlinear continuations of the N normal mode solutions of the corresponding linear system. However, as the energy is increased, these solutions destabilize causing the motion in their vicinity to drift into chaotic domains, thus giving rise to important questions concerning the systems behavior in the thermodynamic limit, where E and N diverge with E=N = constant. One of the open problems in Hamiltonian dynamics, therefore, examines the relation between local (linear) stability properties of simple periodic solutions of Hamiltonian systems, and the more “global” dynamics. In this thesis, after reviewing the main results on these topics for the case of N-particle Fermi-Pasta-Ulam Hamiltonians, I proceed to apply the corresponding methods to a lattice of Hollomon oscillators, which are of interest to applications in problems of nonlinear elasticity.