This course concerns the general description and analysis of the motion of systems of particles acted on by forces. Assuming a basic familiarity with Newton's laws of motion and their application in simple situations, the students will develop the advanced techniques necessary for the study of more complicated systems. The students will also consider the beautiful extensions of Newton's equations due to Lagrange and Hamilton, which allow for simplified treatments of many interesting problems and which provide the foundation for the modern understanding of dynamics. The module also includes an introduction to the calculus of variations, which allows the solution of an important class of problems involving the optimisation of integral quantities.
1. Kinematics and dynamics of particles and systems of particles.
2. Calculus of variations (Equations of Lagrange - first and second kind. Least action principle.)
3. Lagrangian formalism: D'Alembert principle vs variational principle
5. Hamiltonian formalism. Canonical transformations. Noether theorem and symmetries.
The aim of the course is to develop an understanding of how Newton's laws of motion can be used to describe the motion of systems of particles and solid bodies, and how the Lagrangian and Hamiltonian approaches allow use of more general coordinates systems, and how the calculus of variations can be used to solve simple continuous optimization problems.
On completion of the course, students should be able to:
* model simple mechanical systems, using Lagrange's equations;
* analyse the dynamics of systems near equilibrium; find the normal modes of oscillation;
* relate the Hamiltonian and Lagrangian approaches;
* recognise and make use of conserved quantities;
* find the Euler-Lagrange equation associated with simple variational problems.
1. Kinematics and dynamics of particles and systems of particles
Review of Newton's laws; centre of mass; basic kinematic quantities: momentum, angular momentum and kinetic energy; circular motion; 2-body problem; conservation laws; reduction to centre of mass frame.
2. Calculus of variations
Examples of variational problems; derivation of Euler-Langrange equations; natural boundary conditions; constrained systems; examples.
3. Lagrangian formulation of mechanics
Lagrange's equations and their equivalence to Newton's equations, generalized coordinates; constraints; Kepler's law.
Simple harmonic oscillations; coupled harmonic oscillators; normal modes.
5. Hamiltonian formulation
Hamilton's equations, equivalence with Lagrangian formulation; conserved quantities;Poisson brackets, canonical transformations, Liouville's theorems.
- Module Supervisor: Chris Antonopoulos