Lagrangian Dynamics (U01370)

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

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

? 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 Lagrangian Dynamics 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 (PHY-3-TensFlds), Mathematical Physics 3 (PHY-3-MathPhys3). Students wishing to take both Lagrangian Dynamics and Tensors & Fields should enrol for Mathematical Physics 3 (PHY-3-MathPhys3).

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 the Newtonian dynamics of a system of particles
2)Understand virtual displacements, constraints, generalised coordinates/ velocities / forces / momenta; discuss the derivation of the Euler Lagrange equations using virtual displacements
3)Apply the Lagrangian technique to solve a large range of problems in dynamics (the ethereal expression "a large range of problems" may be disambiguated by reference to the tutorial sheets and previous examinations)
4)Understand and apply the calculus of variations, discuss the derivation of the Euler Lagrange equations for constrained systems and thus appreciate Hamilton's principle as the embodiment of Lagrangian dynamics
5)Understand all of: ignorable coordinates / the origin of conservation laws, Lagrangian for a charged particle in an EM field, canonical versus mechanical momentum, allowed changes in the Lagrangian
6)Appreciate the Lagrangian for a relativistic charged particle
7)Derive the conservation of linear (angular) momentum from the homogeneity (isotropy) of space, appreciate the relation between symmetries and conservation laws
8)Define the Hamiltonian by Legendre transformation, derive Hamilton's equations of motion and apply them to simple problems; define / evaluate Poisson brackets, appreciate the connection with Quantum Mechanics
9)Understand rotating frames and the Eulerian approach to rigid body motion, and analyse torque-free motion; understand the Lagrangian formulation of the symmetric top, derive the equations of motion and conservation laws, understand nutation, precession and sleeping
10)Understand and apply small oscillation theory in the Lagrangian formulation
11)Apply all of the above to unseen problems in each formulation of classical dynamics