640-355 Classical and Modern Mechanics

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Objectives:

By the end of this course the student should:

• have a detailed comprehension of the canonical Lagrangian/Hamiltonian formulation of classical mechanics;

• be able to apply it to the analysis of simple systems;

• appreciate the new features that arise in non-linear systems, ranging from the ordered behaviour represented by 'solitons' to the chaotic behaviour displayed by many non-linear classical systems.

Content:

Review of Newtonian mechanics for a system of particles, special relativistic mechanics, non-inertial frames. Review of Lagrangian formulation, and its expression in terms of variational principles; applications to include the general motion of a rigid body. Review of Hamiltonian formulation; Poisson brackets, phase space and Liouville's theorem, Poincaré mapping. Canonical transformations: formulation, generators of infinitesimal transformations, Noether's theorem, the Hamilton-Jacobi method, action and angle variables. Integrable nonlinear systems: pendulum, 2-body Kepler problem, free nonlinear oscillators, limit cycles, separable systems. Non-integrable nonlinear systems: Approximation methods: perturbation theory, adiabatic invariants; chaotic motion in Hamiltonian systems: coupled and driven nonlinear oscillators, the 3-body central force problem; chaotic motion in dissipative systems: the logistic map, Lorenz system, strange attractors; overview of occurrence of deterministic chaos in various areas of physics; criteria for chaotic motion: KAM theorem, Lyapunov exponents. Mechanics of continuous systems: integrable systems, solitons; non-integrable systems, fluid turbulence.

Assessment:

Prescribed Texts:

• Goldstein H, Classical Mechanics, 2nd ed. 1980 Addison-Wesley.