# Mechanics 23

## Unit aims

- To introduce variational principles in mechanics.
- To introduce Lagrangian and Hamiltonian mechanics and their applications.
- To provide a foundation for further study in mathematical physics.

## Unit description

In Newtonian mechanics, the trajectory of a particle is governed by the second-order differential equation F = ma. An equivalent formulation, due to Maupertuis, Euler and Lagrange, determines the particle's trajectory as that path which minimises (or, more generally, renders stationary) a certain quantity called the action. The mathematics which links these two formulations (which at first seem so strikingly different) is the calculus of variations.

The known fundamental laws of physics (e.g., Maxwell's equations for electricity and magnetism, the equations of special and general relativity, and the laws of quantum mechanics) can be formulated in terms of variational principles, and indeed find their simplest expression in this way. The principle of least action in classical mechanics is conceptually one of the simplest, and historically one of the first such examples.

The course covers the principle of least action, the calculus of variations, Lagrangian mechanics, the relation between symmetry and conservation laws, and the theory of small oscillations. The last part of the course is an introduction to Hamiltonian mechanics, including Poisson brackets, canonical transformations. Hamilton-Jacobi theory and some qualitative results.

## Relation to other units

This unit is a more advanced version of the Level 2 unit, Mechanics 2. The lectures for Mechanics 2 and Mechanics 23 are the same, but the problem sheets and examination questions for Mechanics 23 are more challenging. Students may NOT take both Mechanics 2 and Mechanics 23.

This unit develops the mechanics met in the first year from a more general and powerful point of view. Lagrangian and Hamiltonian methods are used in many areas of Mathematical Physics. Familiariaty with these concepts is helpful for Quantum Mechanics, Quantum Chaos, Quantum Information Theory, Statistical Mechanics and General Relativity. Variational calculus, which forms part of the unit, is an important mathematical idea in general, and is relevant to Control Theory and to Optimisation.

## Learning objectives

At the end of the unit the student should:

- understand the notions of configuration space, generalised coordinates and phase space in mechanics
- be able to obtain the Euler-Lagrange equations from a variational principle
- understand the relation between Lagrange's equations and Newton's laws
- be able to use Lagrange's equations to solve complex dynamical problems
- be able to calculate the normal modes and characteristic frequencies of linear mechanical systems
- be able to obtain the Hamiltonian formulation of a mechanical system
- understand Poisson brackets
- understand canonical transformations

## Transferable skills

- Use of mathematical methods to describe "real world" systems
- Development of problem-solving and analytical skills, assimilation and use of complex and novel ideas
- Mathematical skills: Knowledge of the calculus of variations; an understanding of the importance of variational principles in physical theory; analysis of complex problems in mechanics; analysis of linear systems (normal modes, characteristic frequencies)

## Syllabus

0. Introduction

1. Calculus of variations [2 weeks]

- Euler-Lagrange equations in one and more dimensions. Alternative form. Examples: brachistochrone, Fermat's principle.

2. Lagrangian mechanics [3 weeks]

- Principle of least action and Lagrange's equations. Generalised coordinates. Constraints. Derivation of Lagrange's equations from Newton's laws. Conserved quantities (generalised energy, generalised momenta, Noether's theorem). Examples, including spherical pendulum.

3. Small oscillations [2 weeks]

- Normal modes. Stability of equilibria. Examples.

4. Rigid bodies [1.5 week]

- Angular velocity. Inertia tensor. Rigid bodies and Lagrangian mechanics.

5. Hamiltonian mechanics [2.5 weeks]

- Hamilton's equations. Phase space. Conservation laws and Poisson brackets. Canonical transformations. Action-angle variables. Chaos.

There may be minor changes to this syllabus.

## Reading and References

Lecture notes will be provided. Also the later chapters of

*Classical Mechanics*, R. Douglas Gregory, Cambridge University Press (2006)

are especially recommended. Further literature:

*Classical Mechanics*, B. Kibble & Frank H. Berkshire, Imperial College Press (2004)*Analytical Mechanics*, G.R. Fowles & G.L. Cassiday, Saunders College Publishing (1993)- Richard Feynman's lecture on the principle of least action in
*The Feynman Lectures on Physics,*Vol II, Ch 19, R.P. Feynman, R.B. Leighton, and M Sands, Addison-Wesley Publishing (1964) *Classical Mechanics*, 2 ed., H. Goldstein, Addison-Wesley (1980)*Variational Principles in Dynamics and Quantum Theory*, W. Yourgrau and S. Mandelstam, Dover Publications (1968)*The Variational Principles of Mechanics*, 4 ed., C. Lanzcos, Dover Publications (1986)

**Unit code:** MATH31910

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 2 (13-24)

**Lecturers: **Dr Silke Henkes and Dr Sebastian Müller

## Pre-requisites

MATH11007 Calculus 1 and MATH11005 Linear Algebra and Geometry and either MATH11009 Mechanics 1 or PHYS10006 Core Physics I:Mechanics and Matter

## Co-requisites

None

## Methods of teaching

Lectures supported by problem classes and problem and solution sheets.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in May/June

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.