Mathematical Physics 3 (VS2) (U03507)

? Credit Points : 10 ? SCQF Level : 10 ? Acronym : PHY-3-VS2MP3

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.

Entry Requirements

? This course is only available to part year visiting students.

? This course is a variant of the following course : U03233

? Pre-requisites : Year 2 Physics and Mathematics

? Prohibited combinations : Lagrangian Dynamics (pre-2006)

Subject Areas

Home subject area

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

Delivery Information

? Normal year taken : 3rd year

? Delivery Period : Semester 2 (Blocks 3-4)

? Contact Teaching Time : 3 hour(s) per week for 11 weeks

All of the following classes

Type	Day	Start	End	Area
Lecture	Monday	11:10	12:00	KB
Lecture	Thursday	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:

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