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 |
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:
Syllabus
Weeks per topic is approximate at three lectures per week
0. Introduction
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.
At the end of the unit the student should:
understand the relation between Lagrange's equations and Newton's laws
understand Poisson brackets
Transferable Skills:
Lectures supported by problem classes and problem and solution sheets.
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.
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:
are especially recommended.
Further literature: