Description

This module introduces some fundamental concepts in analytical dynamics, and illustrates their applications to relevant problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations, and Hamilton-Jacobi equations. The approach is necessarily mathematical and students are advised to take this optional module only if they have got marks of at least 60% in both PHY1021 Vector Mechanics and PHY1026 Mathematics for Physicists (or in equivalent modules in other departments).

Module Aims

This module will be of interest to students wishing to develop their grasp of theoretical physics. The subject of analytical dynamics provides advanced theoretical developments which prove elegant and versatile in solving dynamical problems.

Intended Learning Outcomes (ILOs)

A student who has passed this module should be able to:

• Module Specific Skills and Knowledge:
  1. solve advanced dynamical problems involving classical particles by applying the Lagrangian and Hamiltonian formulations;
  2. explain the calculus of variations and apply it to the solution of problems;
  3. state the Hamilton-Jacobi equations and apply them to the solution of problems;
  4. describe the relationship between poisson brackets and quantum mechanical commutation relations;
  5. describe the basic concepts of chaos theory and explain how chaos theory may be used in different disciplines;
• Discipline Specific Skills and Knowledge:
  6. formulate mathematical descriptions of physical systems;
• Personal and Key Transferable / Employment Skills and Knowledge:
  7. use mathematics to solve problems.

Syllabus Plan

1. Generalized coordinates. Holonomic and nonholonomic constraints
2. Virtual displacement. D'Alembert's principle
3. The Lagrangian formulation
4. The Hamiltonian formulation
   1. Configuration space; generalized (canonical or conjugate) momentum
   2. Phase space
   3. Legendre transformation
   4. Hamiltonian; Hamilton's equations
   5. Cyclic co-ordinates and conservation theorems
   6. Liouville's theorem
5. Calculus of variations
6. Poisson brackets
   1. Lagrange brackets
   2. Poisson brackets
7. Hamilton-Jacobi equations and action-angle variables
8. The transition to quantum mechanics
9. Nonlinear Dynamical Systems
   1. Chaos and its relevance to mechanics
   2. The stability of non-linear equations
   3. The non-linear oscillator
   4. Phase-Space Methods
   5. The pendulum revisited
   6. Mappings
   7. Characterisation of chaotic systems

Learning and Teaching

Learning Activities and Teaching Methods

Assessment

Resources

The following list is offered as an indication of the type & level of information that students are expected to consult. Further guidance will be provided by the Module Instructor(s).

Core text:

• Gregory R.D. (2006), Classical Mechanics, Cambridge University Press, ISBN 0-521-534097

Supplementary texts:

• Calkin M.G. (1996), Lagrangian and Hamiltonian Mechanics, World Scientific, ISBN 981-02-2672-1
• Englert B.G. (2015), Lectures on Classical Mechanics, World Scientific, ISBN 978-981-4678-44-5
• Fowles G.R. and Cassiday G.L. (2004), Analytical Mechanics (7^th edition), Brooks Cole, ISBN 0-534-49492-7
• Hand L.N. and Finch J.D. (1999), Analytical Mechanics, Cambridge University Press, ISBN 0-521-57572-9
• Kaplan D. and Glass L. (1995), Understanding Nonlinear Dynamics, Springer-Verlag, ISBN 0-387-94440-0
• Kibble T.W.B. (1985), Classical Mechanics (3^rd edition), Longman, ISBN 0-582-45023-3
• Landau L.D. and Lifshitz E.M. (1976), Mechanics (Vol. 1) (3^rd edition), Butterworth-Heinemann, ISBN 978-0-750-62896-9
• Leech J.W. (1965), Classical Mechanics, Methuen and Co., ISBN 0-412-20070-8
• Symon K.R. (1971), Mechanics (3^rd edition), Addison Wesley, ISBN 0-201-07392-7
• ter Haar D. (1971), Elements of Hamiltonian Mechanics, Pergamon Press
• Thornton S.T. and Marion J.B. (2003), Classical Dynamics of Particles and Systems (5^th edition), Tomson, ISBN 0-534-40896-6

ELE:

• http://newton.ex.ac.uk/redirects/ELE/phy2032

Further Information

Prior Knowledge Requirements

Re-assessment

Re-assessment is not available except when required by referral or deferral.

Notes: See Physics Assessment Conventions.

KIS Data Summary

Miscellaneous