Unit information: Mechanics 2 in 2012/13

Unit name	Mechanics 2
Unit code	MATH21900
Credit points	20
Level of study	I/5
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Dr. Muller
Open unit status	Not open
Pre-requisites	MATH11200, MATH11002 and MATH11003
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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, at least, 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 on eof the first such examples. The course covers the principle of least action, the calculus of variations, the derivation of Lagrangian mechanics, and the relation between symmetry and conservation laws. Hamiltonian mechanics is introduced with a treatment of Poisson brackets and liouville's theorem. Applications will include the theory of small oscillations and rigid- body dynamics.

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

Weeks per topic is approximate at three lectures 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 quantitities (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 tensors. Euler's equations.
5. Hamiltonian mechanics [3weeks]. 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 develops the mechanics met in the first year from a more general and powerful point of view. There is a level 3 version, Mechanics 23. Students may NOT take both Mechanics 2 and Mechanics 23.

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 classes and problem and solution sheets.

Assessment Information

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

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)