Nonholonomic dynamical systems : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematical Physics at Massey University

Abstract

The dynamics of mechanical systems subject to nonholonomic (i.e. non-integrable velocity) constraints is only poorly understood. It is known that (i) they preserve energy and, (ii) they are reversible. In this thesis I explore the conjecture that (i) and (ii) are the only general features of the entire class. The discovery of dissipative orbits, ones that behave differently as t→ +∞ and t → -∞, would strongly support this conjecture. This dissipation can appear in various forms, e.g. sinks (attractors) or sources (repellers) in the phase space, but in every form the dynamics have the property that the forwards time orbit occupies a different region of the phase space than the reverse time orbit. In nonholonomic dynamical systems that are reversible and possess an integral, theory predicts that near the fixed set of a reversing symmetry, e.g. R : p→ -p with fixed set Fix(R) = {(q,p) : p = 0}. no dissipation can occur. If the system can be integrated analytically, then all the orbits are quasi-pcriodic and even away from the fixed set of any reversing symmetries, dissipation cannot occur. But, if the system cannot be integrated analytically, then away from the fixed set of any reversing symmetries, dissipative orbits can exist. The minimum dimension needed for a nonholonomic system is 6. So, in this thesis I study the simplest class of nonholonomic dynamical systems that are reversible with an integral, namely the contact particle in R 3 . I search for evidence of dissipative behaviour in this class of systems by taking a known contact particle system that can be integrated analytically, such as the harmonic oscillator, where no dissipation can occur and calculating (numerically and analytically) the dynamics of its orbits. Then I perturb the system so that it cannot be integrated analytically and search for orbits that exhibit the dissipative behaviour described above away from the fixed set of the reversing symmetries of the system. To achieve this I implemented a semi-explicit reversible integrator in C to integrate the system forwards (or backwards when desired) in time from an initial point. The C code interacts with MATLAB via the "mex" interface to make use of MATLAB's graphing facilities, which I used to plot the forwards and backwards orbits in blue and red respectively. This allows the orbits to be observed and any dissipative behaviour should become immediately apparent as the orbits will cover different portions of the phase space if dissipation occurs. The phase space of the system is actually R 3 , which is beyond my capabilities to visualise, but it can be reduced to R 3 , as I have done, through the use of the integral, the nonholonomic constraint and a Poincaré section.