| Unit name | Mechanics 2 |
|---|---|
| Unit code | MATH21900 |
| Credit points | 20 |
| Level of study | I/5 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Dr. Mike Blake |
| Open unit status | Not open |
| Pre-requisites |
MATH10012 ODEs, Curves and Dynamics (or MATH11007 Calculus 1 and PHYS10006 Core Physics 1) and MATH11005 Linear Algebra and Geometry |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
Unit Aims
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.
Relation to Other Units
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:
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)
Lectures supported by problem classes and problem and solution sheets.
Recommended
Further
