# PX267 Hamiltonian Mechanics

## Lecturer: James Lloyd-Hughes

## Weighting: 7.5 CATS

This module introduces the Hamiltonian formulation of classical mechanics. This elegant theory has provided the natural framework for several important developments in theoretical physics including quantum mechanics. The module starts by covering the general "spirit" of the theory and then goes on to introduce the details. The module uses a lot of examples. Many of these should be familiar from earlier studies of mechanics while others, which would be much harder to deal with using traditional techniques, can be dealt with quite easily using the language and methods of Hamiltonian mechanics.

**Aims:**

To revise the key elements of Newtonian mechanics and use this to motivate and then develop Lagrangian and Hamiltonian mechanics

**Objectives:**

At the end of the module, you should

- Understand the significance of the Lagrangian. You should be able to derive and solve the Euler-Lagrange equations for simple models.
- (Working from the Lagrangian) be able to find the canonical momenta and to construct the Hamiltonian function
- Be able to derive and solve Hamilton's equations for simple systems
- Appreciate the role of (and relations between) constraints, conserved quantities and generalised coordinates

**Syllabus:**

1. Introduction. Analogy with Optics and constructive interference; principle of least action; examples of L: T-V, -mc2/γ

2. Euler Lagrange Equations. 1-d trajectory, T-V case, worked examples; T+V as a constant of the motion; multiple coordinates with examples

3. Generalised Coordinates and Canonical Momenta. Polar coordinates; angular momentum; moment of inertia of rigid bodies; treatment of constraints; examples

4. Symmetry and Conservation Laws

5. Hamiltonian Formulation. Hamilton's Equations, phase space, examples

6. Normal Modes and Small Oscillations. Inertial and stiffness matrices, diatomic and Triatomic molecules

**Commitment:** about 18 Lectures

**Assessment:** 1 hour examination

This module has a home page.

**Recommended Text:** A good text going well beyond the module is H Goldstein, Classical Mechanics; A helpful reference for the beginning of the module is: Feynmann, Leighton & Sands, The Feynmann Lectures on Physics, Vol 2, Chapter 19

**Leads from:** PX148 Classical Mechanics and Relativity