ACM30010

This course introduces students to the principles of mechanics in a mathematical framework and establishes a bridge to the principles of modern physics. The module begins with an introduction to Lagrangian mechanics which enables Newtonian mechanics to be distilled in its purest form. The powerful theory of Calculus of Variations is introduced and several crucial examples are studied, such as the brachistochrone. The course continues by developing Hamiltonian mechanics, a re-formulation of classical mechanics that was invented by William Rowan Hamilton, Ireland's greatest Mathematician.

Course Outline (sample):

1. Introduction to Analytical Mechanics. Generalised Co-ordinates and Generalised Forces in the Mechanical Description of a Single Particle. Degrees of Freedom. Equations of motion. Lagrange equations. Special case of conservative forces: Lagrangian. Ignorable coordinates (e.g., central force problem). Example: the simple pendulum.

2. The Lagrange Equations for a system of particles. Special case of conservative forces: Lagrangian of the N-particle system. Examples: Double pendulum and small oscillations thereof: normal modes and normal frequencies of oscillation.

3. Small Oscillations. General treatment in the conservative-force case. Lagrangian for small oscillations. Normal modes and normal frequencies of oscillations. Diagonalisation of the Lagrangian. Examples: different configurations of systems of masses connected by springs.

4. Hamilton's Principle. Historical context. Minimisation of the action integral. Calculus of Variations. Examples: shortest-length paths (geodesics) on the plane; the brachistochrone; geodesics in spherical surfaces (routes of international airplanes). Extension to N-dimensional paths: the Euler-Lagrange equations. Ignorable coordinates. Hamilton's Principle for mechanical systems.

5. Rigid Body Motion and Euler-Lagrange equations. Degrees of freedom. The angular velocity. The equations of motion and the inertia tensor. Application: freely rotating bodies. Lagrangian formulation. Holonomic constraints. Kinetic energy. Euler angles.

6. The Hamilton Equations. Motivation as the basis for statistical mechanics and quantum mechanics. The Hamiltonian. Conjugate variables. Examples for particles constrained to move on given surfaces.

7* (Extra Material). Classical relativistic particles immersed in a given electromagnetic field: Hamiltonian and Lagrangian approaches. Introduction to the quantum theory through Feynman's path-integral formulation. Example: the double-slit experiment for electron scattering/diffraction.