Unit information: Mechanics 23 in 2013/14

Unit name	Mechanics 23
Unit code	MATH31910
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Dr. Muller
Open unit status	Not open
Pre-requisites	MATH11200 Mechanics 1, MATH20900 Calculus 2, MATH20101 Ordinary Differential Equations 2
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

The unit assumes knowledge of elementary mechanics at the level of the unit Mechanics 1 (MATH11200), and develops the material in two ways: to deal with more advanced mechanical systems in three dimensions (small osscilliations; rigid-body mechanics), and to to introduce more sophisticated mathematical formulations (variational principles, Hamiltonian mechanics) which pave the way for quantum mechanics and for the thory of dynamical systems. The unit is based on the same course of lectures as the Level 1 unit MATH21900 Mechanics 2. But this H level unit will have a different examination from the level I unit, testing students' understanding at the higher level appropriate for level H, and using the mathematical techniques learnt in Level I units.

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.

Syllabus

Please note that this course contains 33 lectures at 3 per week.

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 [1.5 weeks] Normal modes. Stability of equilibria. Examples.

4. Rigid bodies [1.5 weeks] Angular velocity. Inertia tensor. Euler's equations.

5. Hamiltonian mechanics [3 weeks] Hamilton's equations. Phase space. Conservation laws and Poisson brackets. Liouville's theorem. Canonical transformations. Action-angle variables. Chaos.

There may be minor changes to this syllabus.

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.

Intended Learning Outcomes

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

• understand Liouville's theorem on conservation of phase volume and to appreciate some of its consequences.

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)

Teaching Information

Lectures supported by problem and solution sheets.

Assessment Information

The assessment mark for Mechanics 23 is calculated from a 2 ½-hour written examination in April consisting consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted be used in this examination.

Reading and References

Lecture notes will be provided. (See http://www.maths.bris.ac.uk/~maxsm/mechnotes.pdf for last year's version.)

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)

• Mechanics, 3 ed, L.D. Landau & E.M Lifschitz, Pergamon (1976)

• Mathematical Methods of Classical Mechanics, V.I. Arnold, Springer-Verlag (1978)

• 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)