Unit name | Mechanics 23 |
---|---|

Unit code | MATH31910 |

Credit points | 20 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Dr. Mike Blake |

Open unit status | Not open |

Pre-requisites |
MATH11007 Calculus 1 and MATH11009 Mechanics 1 (or MATH10012 ODEs, Curves and Dynamics) and MATH11005 Linear Algebra and Geometry. Core Physics A can be used in replace of Mechanics 1. |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**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 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)

Lectures supported by problem classes and problem and solution sheets.

- 90% Exam
- 10% Coursework

**Recommended**

- The later chapters of R. Douglas Gregory, Classical Mechanics, Cambridge University Press, 2006 will be provided

**Further**

- Richard P. Feynman, Robert B. Leighton, and M Sands,
*The Principle of Least Action*in*The Feynman Lectures on Physics,*Vol II, Ch 19, Addison-Wesley Publishing, 1964 - Grant R. Fowles and George L. Cassiday,
*Analytical Mechanics*, Saunders College Publishing, 1993 - Herbert Goldstein,
*Classical Mechanics*, 2 ed., Addison-Wesley, 1980 - T. W. B. Kibble and F. H. Berkshire.
*Classical Mechanics*, Imperial College Press, 2004 - Cornelius Lanzcos,
*The Variational Principles of Mechanics*, 4 ed., Dover Publications, 1986 - Wolfgang Yourgrau and Stanley Mandelstam,
*Variational Principles in Dynamics and Quantum Theory*, Dover Publications, 1968