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.

By the end of the module the student should be able to:

• analyse a dynamical system’s phase-space behaviour, in the Hamiltonian case or near-Hamiltonian case, including smooth phase flow, fixed points and limit cycles and in more detail elliptic and hyperbolic points and separatrices;
• construct action-angle variables and use them to give geometric representations and allow averaging and separation of fast and slow variables;
• systematically exploit separations of time scales – to produce averaged equations of motion and define and deduce consequences of adiabatic invariants;
• understand the concept of a nonlinear resonance, illustrated by the example of the nonlinear pendulum;
• analyse simple cases of nonlinear resonance; understand the definition of, and be able to calculate (in simple cases), Lyapunov exponents; analyse maps and understand their origin and relation to mixing and ergodicity.