BSc Physics with Theoretical Physics

The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

Lagrangian Dynamics

To introduce the Lagrangian and Hamiltonian formulations of classical mechanics. To develop the knowledge and skills required to solve a variety of dynamical problems involving more than one degree of freedom.

On completion successful students will be able to:

1. Choose an appropriate set of generalised coordinates to describe a dynamical system and obtain its Lagrangian in terms of those coordinates and the associated 'velocities'. Derive and solve the corresponding equations of motion. Treat small oscillations as an eigenvalue problem.

2. Apply a variational principle to solve simple problems involving constraints.

3. Appreciate symmetries and how they manifest themselves in terms of constants of the motion.

4. Obtain generalised momenta and thus the Hamiltonian of a dynamical system. Derive and solve the equations of motion in Hamiltonian form.
 

1.  Introduction

Review of Newtonian mechanics:  internal forces, external forces, forces of constraint.  Rotational problems and polar coordinates.
Conservation laws and conservative systems.
Partial derivatives.

2.  Lagrangian Dynamics

The energy method plus other conservation laws.
The Lagrangian and Lagrange’s equation.
Small oscillations and normal modes.

3.  Calculus of Variations

Functional minimization.
The Euler-Lagrange equations.
Constrained variation.
Hamilton’s principle of least action.
Lagrangian dynamics.

4.  The Hamiltonian Formalism

Legendre transformations.
Generalized momenta, the Hamiltonian and Hamilton's equations.
Phase space.  Liouville’s theorem

5.  Symmetries and Conservation Laws

Generators of transformations.
Poisson brackets.
Symmetries of the Lagrangian produce constants of motion. Noether’s theorem.

6.  Normal Modes from Matrices

Normal modes from symmetries.
Review of mathematics of matrices:  eigenvalues and eigenvectors.
Diagonalizing a matrix using its eigenvectors.
Small oscillations as eigenvalue problems.

7.  Special Topics

Lagrangian for charged particle moving in electric and magnetic fields.
Continuous systems: the Lagrangian Density.

Model answers will be issued within one week of issuing each example sheet. Informal Q&A sessions will be organised to allow students to clarify any questions on the lecture material or on the model answers.

Kibble, T.W.B. & Berkshire, F.H. Classical Mechanics, 5th edition (Longman)
Goldstein, H., Poole, C. & Safko, J. Classical Mechanics, 3rd edition (Addison-Wesley)
Landau, L.D. and Lifshiftz, E.M. Mechanics, 3rd edition (Pergamon Press)