Mathematical Physics 3 (U03233)

? Credit Points : 20 ? SCQF Level : 10 ? Acronym : PHY-3-MathPhys3

Lagrangian Dynamics: The principles of classical dynamics, in the Newtonian formulation, are expressed in terms of (vectorial) equations of motion. These principles are recapitulated and extended to cover systems of many particles. The laws of dynamics are then reformulated in the Lagrangian framework, in which a scalar quantity (the Lagrangian) takes centre stage. The equations of motion then follow by differentiation, and can be obtained directly in terms of whatever generalised coordinates suit the problem at hand. These ideas are encapsulated in Hamilton's principle, a statement that the motion of any classical system is such as to extremise the value of a certain integral. The laws of mechanics are then obtained by a method known as the calculus of variations. As a problem-solving tool, the Lagrangian approach is especially useful in dealing with constrained systems, including (for example) rotating rigid bodies, and one aim of the course is to gain proficiency in such methods. At the same time, we examine the conceptual content of the theory, which reveals the deep connection between symmetries and conservation laws in physics. Hamilton's formulation of classical dynamics (Hamiltonian Dynamics) is introduced, and some of its consequences and applications are explored.

Tensors & Fields. This course provides an introduction to tensors and their uses in physics. Topics covered include: vectors, bases, determinants and the index notation; the general theory of Cartesian tensors; rotation and reflection tensors; various applications including elasticity theory - stress and strain tensors; orthogonal curvilinear coordinates, grad, div and curl; the divergence and Stokes' theorem, Laplacians; delta functions; applications to electromagnetism: scalar potential theory, e.g. Gauss's Law, dipoles, multipole expansions, solutions of Laplace's & Poisson's equations.

Entry Requirements

? Pre-requisites : Foundations of Mathematical Physics (PHY-2-FoMP) or Principles of Mathematical Physics (PHY-2-PoMP) and Methods of Applied Mathematics (MAT-2-MAM). Students intending on taking Mathematical Physics 3 in Junior Honours must have obtained a minimum grade of 'C' in Foundations of Mathematical Physics (PHY-2-FoMP) or a minimum average grade of 'C' in Principles of Mathematical Physics (PHY-2-PoMP) and Methods of Applied Mathematics (MAT-2-MAM).

? Prohibited combinations : Tensors & Fields (pre-2006) Lagrangian Dynamics (pre-2006)

Variants

? This course has variants for part year visiting students, as follows

Mathematical Physics 3 (VS1) (Semester 1 (Blocks 1-2))
Mathematical Physics 3 (VS2) (Semester 2 (Blocks 3-4))

Subject Areas

Home subject area

Undergraduate (School of Physics), (School of Physics, Schedule Q)

Delivery Information

? Normal year taken : 3rd year

? Delivery Period : Full Year (Blocks 1-4)

? Contact Teaching Time : 3 hour(s) per week for 22 weeks

First Class Information

Date	Start	End	Room	Area	Additional Information
19/09/2006	11:00	12:00	Lecture Room 5327, JCMB	KB

All of the following classes

Type	Day	Start	End	Area
Lecture	Tuesday	11:10	12:00	KB
Lecture	Friday	11:10	12:00	KB

? Additional Class Information : Workshop/tutorial sessions, as arranged.

Summary of Intended Learning Outcomes

On successful completion of this course a student will be able to:
LD:
1)Understand Newtonian dynamics of a system of particles, virtual displacements, constraints, generalised coordinates/velocities/forces/momenta; discuss derivation of Euler Lagrange equations using virtual displacements
2)Apply the Lagrangian technique to solve a large range of problems in dynamics
3)Understand and apply calculus of variations, discuss derivation of Euler Lagrange equations for constrained systems and thus appreciate Hamilton's principle as the embodiment of Lagrangian dynamics
4)Understand ignorable coordinates/origin of conservation laws, Lagrangian for a charged particle in an EM field, canonical vs mechanical momentum, allowed changes in the Lagrangian
5)Appreciate the Lagrangian for a relativistic charged particle
6)Derive conservation of linear/angular momentum from homogeneity/isotropy; appreciate symmetry-conservation law connection
7)Define the Hamiltonian by Legendre transformation; derive, apply Hamilton's equations; define/evaluate Poisson brackets; appreciate connection with Quantum Mechanics
8)Understand rotating frames, Eulerian approach to rigid body motion; analyse torque-free motion; understand Lagrangian formulation of symmetric top, derive equations of motion, conservation; understand nutation, precession, sleeping
9)Understand and apply small oscillation theory in Lagrangian formulation
10)Apply the above to unseen problems in each formulation of classical dynamics
T&F:
1)be confident with index notation, Einstein convention
2)have knowledge of matrices and determinants and derive vector identities
3)understand meaning and significance of tensors and application to simple physical situations
4)understand and manipulate various orthogonal curvilinear coords
5)be familiar with divergence and Stokes' theorem
6)appreciate Dirac delta fn.
7)understand the meaning of a field and potential
8)solve potential theory problems
9)solve new proble