PHY3149 Analytical Dynamics

2010-2011

Code: PHY3149
Level: 3
Title: Analytical Dynamics
Instructors: Prof. G.P. Srivastava
CATS Credit Value: 10
ECTS Credit Value: 5
Pre-requisites: N/A
Co-requisites: N/A
Duration: M1-M11
Availability: unrestricted
Background Assumed: Relativity I and Vectors (PHY1105) and minimum of 55% in Mathematics for Physicists (PHY1116)
Directed Study Time: 22 lectures
Private Study Time: 78 hours
Assessment Tasks Time: -

Aims

The subject of analytical dynamics provides advanced theoretical developments which prove elegant and versatile in solving dynamical problems. 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.

This module will be of particular interest to students wishing to study the methods of theoretical physics. Its approach is necessarily mathematical and students should note the assumed background specified above.

Intended Learning Outcomes

A student should be able to:

Module Specific Skills

• solve advanced dynamical problems involving classical particles by applying the Lagrangian and Hamiltonian formulations;
• explain the calculus of variations and apply it to the solution of problems;
• state the Hamilton-Jacobi equations and apply them to the solution of problems;
• describe the relationship between poisson brackets and quantum mechanical commutation relations;

Discipline Specific Skills

• formulate problems and solve using mathematical methods;

Personal and Key Skills

• problem solving.

Learning / Teaching Methods

Lectures, self-study problems and discussion

Assignments

Students are regularly given problems to solve as home work. Solutions to such problems are discussed in the class.

Assessment

One 90-minute examination (100%).

Syllabus Plan and Content

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
   1. Stationary values; external values
   2. Euler-Lagrange equations
   3. Hamilton's principle; Hamilton's equations
   4. Principle of least action
6. Poisson brackets
   1. Lagrange brackets
   2. Poisson brackets
   3. Jacobi's identity
7. Hamilton-Jacobi equations
8. The transition to quantum mechanics

Core Text

Gregory R.D. (2006), Classical Mechanics, Cambridge University Press, ISBN 0-521-534097 (UL: 531 GRE)

Supplementary Text(s)

Calkin M.G. (1996), Lagrangian and Hamiltonian Mechanics, World Scientific, ISBN 981-02-2672-1 (UL: 531 CAL)
Fowles G.R. and Cassiday G.L. (2004), Analytical Mechanics (7^th edition), Brooks Cole, ISBN 0-534-49492-7 (UL: 531FOW)
Goldstein H., Poole C. and Safko J. (2002), Classical Mechanics (3^rd edition), Addison Wesley, ISBN 0-201-65702-3 (UL: 531 GOL)
Kibble T.W.B. (1973), Classical Mechanics, McGraw Hill (UL: 531 KIB)
Landau L.D. and Lifshitz E.M. (1976), Mechanics, Pergamon Press (UL: 531 LAN)
Leech J.W. (1965), Classical Mechanics, Methuen and Co. (UL: 531 LEE)
Symon K.R. (1971), Mechanics (3^rd edition), Addison Wesley, ISBN 0-201-07392-7 (UL: 531 SYM)
ter Haar D. (1971), Elements of Hamiltonian Mechanics, Pergamon Press (UL: 531 HAA)
Thornton S.T. and Marion J.B. (2003), Classical Dynamics of Particles and Systems (5^th edition), Tomson, ISBN 0-534-40896-6 (UL: 531.11MAR)

Formative Mechanisms

Students' home work is checked in class, and general difficulties are sorted out. Guidance is provided to make further progress in understanding the course work and in solving problems.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.