Classical Mechanics

Learning Outcomes

In this course students will embark on a voyage of discovery of the deep theoretical principles
that underlie Newtonian and relativistic mechanics, and to appreciate why the laws
of physics are the way they are. They will learn new ways of thinking about the physical
world which allow deeper appreciation of the links between the classical and quantum
regimes.

Armed with the powerful techniques of Lagrangian and Hamiltonian dynamics, and Cartesian
tensors, students will have the tools to simplify complex mechanical problems to their
basic elements. With elegant symmetry principles such as Noether’s theorem they will
understand the deep connection between symmetries of spacetime and conservation laws,
seeing how, for example, Kepler’s second law follows from rotational symmetry and conservation
of angular momentum. They will apply this new understanding to a variety of
physical systems, from coupled oscillators to particles moving in electromagnetic fields.
Finally they will discover how the symmetries of special relativity are most succinctly described
with the language of 4-vectors, and derive the Lorentz group from the Principle of
Relativity.

This course is the basis for all advanced courses in theoretical physics.

OUTLINE
* Dynamical systems – definitions. Constrained systems. Lagrange’s equations.
* Principle of least action. Euler-Lagrange equations.
* Symmetries, conservation laws and Lie groups. Noether’s theorem.
* Oscillations: linearization. The linear chain.
* Hamiltonian formulation. Legendre’s transformation.
* Transformation theory. Canonical transformations. Generating functions. Poisson
brackets.
* Hamilton-Jacobi method. Physical applications: (e.g. wave mechanics and Schr¨odinger’s
equation).
* Special relativity: Kinematics, symmetries and Lagrangian formulation

Prerequisites

For further information see School of Physical & Chemical Sciences .