|Unit level:||Level 2|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
The course unit aims 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.
This course concerns the general description and analysis of the motion of systems particles acted on by forces. Assuming a basic familiarity with Newton's laws of motion and their application in simple situations, we shall develop the advanced techniques necessary for the study of more complicated systems. We shall 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 maximization or minimization of integral quantities. The course is a useful primer to third and fourth level course units in physical applied mathematics.
On completion of this unit successful students will be able to:
- model simple mechanical systems, using Lagrange's equations;
- analyze the dynamics of systems near equilibrium; find the normal modes of oscillation;
- relate the Hamiltonian and Lagrangian approaches;
- recognize and make use of conserved quantities;
- find the Euler-Lagrange equation associated with simple variational problems.
- Other - 20%
- Written exam - 80%
Assessment Further Information
- Coursework (worth 20%) set around the middle of the semester
- End of semester examination (worth 80%).
1. Newtonian Mechanics of 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; cyclic variables; examples. 
4. Potential wells and oscillations
Particle in a potential well; coupled harmonic oscillators; normal modes. 
Hamilton’s equations, equivalence with Lagrangian formulation; equilibria; conserved quantities. 
- Classical Mechanics, by R.D. Gregory, CUP.
- Classical Mechanics, by T.W.B. Kibble. F.H. Berkshire, Addison Wesley
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours