PH4038 Lagrangian and Hamiltonian Dynamics

Further information on which modules are specific to your programme.

Module description

The module covers the foundations of classical mechanics as well as a number of applications in various areas. Starting from the principle of least action, the Lagrangian and Hamiltonian formulations of mechanics are introduced. The module explains the connection between symmetries and conservation laws and shows bridges between classical and quantum mechanics. Applications include the central force problem (orbits and scattering) and coupled oscillators.

Additional information from school

PH4038 - Lagrangian and Hamiltonian Dynamics

Aims & Objectives

To give students a solid grounding and sufficient training in Lagrangian and Hamiltonian techniques in classical mechanics and their applications, including

• the Principle of Least Action as the starting point of Lagrangian mechanics
• traditional applications of Lagrangian mechanics such as mechanical pendulums, planetary motion, collisions and some non-traditional ones
• appreciating the problem-solving power, generality and elegance of Lagrangian and Hamiltonian techniques
• understand the fundamental connection between symmetries and conservation laws (Noether theorem)

Learning Outcomes

By the end of the module, students will have a solid knowledge of the central concepts of Classical Mechanics and will have acquired and trained important problem-solving skills. They will be able to

• establish the Lagrangian, and to derive and solve the equations of motions for many systems subject to the Principle of Least Action
• calculate conserved quantities from symmetries
• calculate the Hamiltonian and establish Hamilton’s equations
• be familiar with canonical transformations and Hamilton-Jacobi theory
• understand the concept of phase space and the conservation of phase-space density (Liouville's theorem)
• acquire a deep knowledge of the Hamiltonian formalism that is crucial for the formulation and understanding of quantum mechanics

Synopsis

Review of Newtonian mechanics. Functionals and functional derivatives, Euler-Lagrange equations. Lagrangian, Principle of Least Action, symmetries and conservation laws: energy, momentum, angular momentum, centre of mass. Central forces and orbits, Kepler problem (planetary motion), scattering problems, Rutherford scattering. Hamiltonian formalism, canonical momenta, Hamilton’s equations, Poisson brackets, canonical transformations. Application to circuit electrodynamics, filters and transmission lines, classical field theory. Canonical mechanics: symmetries and conservation laws, Noether’s theorem, Liouville’s theorem, Hamilton-Jacobi formalism.

Additional information on continuous assessment etc.

Please note that the definitive comments on continuous assessment will be communicated within the module. This section is intended to give an indication of the likely breakdown and timing of the continuous assessment.

This module is typically taken in JH by theoretical physicists, and in SH by those doing an MPhys in other degree programmes in the School. Five tutorial sheets will be issued over the semester in two week intervals. They contain questions that will deepen the understanding of the current topics in the lectures, and they are required to be handed in for marking. This accounts for 25% of the module mark. Tutorials take the form of “whole class” tutorials where the solutions and underlying physics and problem-solving strategies can be discussed.

Accreditation Matters

This module may not contain material that is part of the IOP “Core of Physics”, but does contribute to the wider and deeper learning expected in an accredited degree programme. The skills developed in this module, and others, contribute towards the requirements of the IOP “Graduate Skill Base”.

Recommended Books

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