There are physical situations where to a very high accuracy energy is conserved. Examples are planetary (and asteroid) motion in the solar system and some aspects of charged particle motion in particle physics accelerators (before collisions occur!). Simple examples of such Hamiltonian systems are the focus of this module. This module provides an introduction to chaos and dynamical systems in conservative (or almost conservative) systems. Initially the module introduces the landscape of typical non-chaotic dynamical systems in two-dimensional phase-space: elliptic and hyperbolic fixed points, the separatrix, and limit cycles for weakly dissipative systems. Action-angle variables are introduced and tori are shown to be the natural higher dimensional objects for conservative systems. Separation and resonance of time scales forms the middle part of the module, where highly counter-intuitive phenomena are illustrated (pendula oscillating about their maximum of potential energy, for example). The nonlinear pendulum is used as the main example here and the origin of chaos is isolated as the separatrix. Finally maps are constructed which show the nature of chaos clearly: Lyapunov exponents showing the sensitivity to initial conditions; ergodicty and mixing illustrating the basis of equal a priori probabilities in statistical mechanics and the decay of correlations with time in physical variables. |