This module is not currently running in 2024 to 2025.
This module will present a new perspective on Newton's familiar laws of motion. First we introduce variational calculus with applications such as finding the paths of shortest distance. This will lead us to the principle of least action from which we can derive Newton's law for conservative forces. We will also learn how symmetries lead to constants of motion. We then derive Hamilton's equations and discuss their underlying structures. The formalisms we introduce in this module form the basis for all of fundamental modern physics, from electromagnetism and general relativity, to the standard model of particle physics and string theory.
Indicative syllabus:
Review of Newton mechanics: Newton's law; harmonic and anharmonic oscillators (closed and unbound orbits, turning points); Kepler problem: energy and angular momentum conservation
Lagrangian Mechanics: Introdution to variational calculus with simple applications (shortest path - geodesic, soap film, brachistochrone problem); principle of least action: Euler-Lagrange equations (Newtonian mechanics with conservative forces); constraints and generalised coordinates (particle on a hoop, double pendulum, normal modes); Noether's theorem (energy and angular momentum conservation)
Hamiltonian Dynamics: Hamilton's equations; Legendre transform; Hamiltonian phase space (harmonic oscillator, anharmonic oscillators and the mathematical pendulum); Liouville's theorem; Poisson brackets.
Total contact hours: 42
Private study hours: 108
Total study hours: 150
80% Examination, 20% Coursework
Douglas Gregory, "Classical Mechanics", Cambridge University Press 2006.
Herbert Goldstein, Charles P Poole; John L Safko; "Classical mechanics", Pearson/Addison Wesley, Third edition, 2002.
Patrick Hamill, "A student's guide to Lagrangians and Hamiltonians", Cambridge University Press 2014.
Emmanuele DiBenedetto, "Classical Mechanics Theory and Mathematical Modeling", Boston, MA : Birkha¨user Boston : Imprint: Birkha¨user 2011.
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes.
On successfully completing the module students will be able to:
1 demonstrate knowledge and critical understanding of the well-established principles within Lagrangian and Hamiltonian formulations of Newtonian mechanics, particularly
the dynamics of conservative systems;
2 demonstrate the capability to use a range of established techniques and a reasonable level of skill in calculation and manipulation of the material to solve problems in the
following areas: variational calculus, use of generalised coordinates, application of constraints, Euler-Lagrange equations, conserved quantities, Hamiltonian formulation,
the Legendre Transform, interpretation of phase portraits, use of Poisson brackets;
3 apply the concepts and principles in basic Lagrangian and Hamiltonian dynamics in well-defined contexts beyond those in which they were first studied, showing the
ability to evaluate critically the appropriateness of different tools and techniques.
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