A global, thermodynamic potential approach to complex systems and deterministic chaos
Peter Scott Geissler, Robin Shepherd · 1993
Engineers need new mathematical tools for the systematic study of chaotic phenomena because of the non-integrability of the equations of motion. This thesis is aimed at developing conceptual and mathematical tools suitable for such analysis. It is shown that integral formulations of mechanics based upon state hold great promise for the study of chaos. The validity of Hamilton's Principle of Least Action is demonstrated for the chaotic dynamics exhibited by a harmonically driven, damped nonlinear pendulum constrained to move in a vertical plane. The state-function character of the action integral is also established for this dynamical system. It is further shown that the chaotic trajectory selected, naturally, by the dynamical system minimizes the action integral and that Hamilton's principle yields a global, rather than merely a local, minimum. A Raleigh-Ritz procedure based upon a minimization of Hamilton's action integral is shown to converge rapidly. The validity of two central theorems in classical mechanics, Lagrange's relation and the Hamilton-Jacobi equation is shown to hold for chaotic dynamics. It is further demonstrated how the state-function character of the action integral imposes a constraint, in addition to the equation of motion, upon the possible behavior of dynamical systems. New results are obtained by utilizing the state-function character of the action integral, together with the equation of motion. The fundamental mathematical problem encountered in applying Hamilton's Principle of Least Action to the calculation of chaotic trajectories is the requirement that the endpoint of the trial functions must be specified a priori. The feasibility of computer generated random walk (e.g. Monte Carlo) calculations is examined in light of this tethering condition. The tethering condition is shown to drastically reduce the efficacy of Monte Carlo calculations. A new, variational principle is derived which is similar to Hamilton's Principle of Least Action, but which eliminates the requirement for the a priori specification of the endpoint of the trajectory in favor of natural boundary conditions.