Course Overview: This course builds on the Prelims Dynamics course, recasting Newtonian mechanics in the Lagrangian and Hamiltonian formalisms. As well as being elegant and computationally useful, these formulations of classical mechanics give important insights into symmetries and conservation laws, and are the language used to describe all modern theories of physics.

Learning Outcomes: Students will be able to demonstrate knowledge and understanding of the Lagrangian and Hamiltonian formalisms. They will understand how symmetries and conserved quantities are described in this language, and be able to apply the ideas developed to small oscillations around equilibria, rigid body motion and some other elementary systems.

Course Synopsis: Review of Newtonian mechanics. Generalized coordinates. The principle of least action. Constraints. Symmetries and Noether's Theorem. Examples with simple systems.

Equilibria. Small oscillations about a stable equilibrium and normal modes, with examples.

Rigid bodies. Angular velocity, angular momentum and the inertia tensor. Euler's equations and tops. Euler angles and \(SO(3)\).

Legendre transformations and the Hamiltonian. Phase space and its geometry. Poisson brackets. Canonical transformations. Liouville's theorem. The Hamilton-Jacobi equation.