I am currently teaching Mathematical Methods for Natural Sciences. In the past I have supervised:

• Lent 2009: Mathematical Methods
• Michaelmas 2008: Mathematical Methods, Dynamical Systems
• Easter 2008: Mathematical Methods, Dynamical Systems revision
• Lent 2008: Newtonian Dynamics, Mathematical Methods
• Michaelmas 2007: Dynamical Systems, Mathematical Methods
• Easter 2007: Classical Dynamics revision, Dynamical Systems revision
• Lent 2007: Newtonian Dynamics, Dynamical Systems
• Michaelmas 2006: Classical Dynamics

If you're being supervised by me: Hand in work by 5pm the day before the supervision and I will mark it and return it to you the next day. You can hand in work to my house (26 Portugal Street) or to the 'T' pigeonhole in DAMTP in the Centre for Mathematical Sciences. Supervisions for Natural Sciences are in the Old Post Room, Whewell's Court, Trinity College. I will normally hand out model solutions after the supervision.

Mathematical Methods (IANST)

Example sheets: Michaelmas 1, 2 and Lent 1, 2, 3.
Revision: Michaelmas, Lent, Easter. Guide to stationary points.

The Natural Sciences mathematics course covers all of the mathematics that you'll need to know to take any of the courses in the second year. Mathematics is vital to science (and science is vital to mathematics, although many mathematicians would disagree) and it's necessary to be familiar with it if you want to study science at anything other than the most basic level.

The course is split very roughly into the following ten sections:

• Vectors
• Complex numbers
• Calculus (integration and differentiation)
• Probability
• Ordinary differential equations
• Partial differentiation
• Area and volume integrals
• Scalar and vector fields
• Matrix methods
• Fourier series

The first four are lectured in Michaelmas, the next four in Lent and the final two in Easter.

Dynamical Systems (IIC)

Example sheets: 1, 2, 3, 4.
Revision: revision sheet

In this course you study the properties of time-dependent (i.e. dynamic) systems in the form of differential equations or difference equations. The kinds of systems you will study are nonlinear, which generally means that there is no analytic solution. Despite this it is still possible to say a lot about the solutions to the equations, especially if you concentrate on the long-time behaviour (i.e. what the system `settles down to') and ignore transients. You can ask questions like:

• Are there stationary states, and if so are they stable?
• Are there time-periodic solutions, or solutions which eventually become time-periodic?
• Are there conserved quantities, or almost-conserved quantities?
• How does the behaviour of the system change as we vary parameters in the equations?

Many of the methods taught in the course are geometric, rather than analytic. Drawing pictures (phase portraits) of the solutions gives you a good qualitative grasp of what's going on, even when you can't write down an analytic solution.

At the end of the course you will study the phenomenon of chaos. Systems which display chaos have extreme sensitivity to the initial conditions: two solutions which initially look similar can diverge wildly sometime later. It is chaos that makes it so hard to give accurate weather forecasts, and it is important in the study of turbulent fluid flow.

Newtonian Dynamics (IA)

Example sheets: 1, 2, 3, 4.

This course will be your first taster of theoretical physics - the methods that you learn here will be used later if you go on to study fluid dynamics, electrodynamics, quantum mechanics or general relativity. But first, you need to master the basics!

You will meet Newton's three laws of motion. These are the basis for much of classical physics, and you can essentially treat them as a set of axioms for motion. You will apply them in different contexts, for example to gravitation and to particles moving in electric and magnetic fields. You will also look at the effects of friction and dissipation, and learn to make use of concepts such as energy, momentum, angular momentum, work and power. Towards the end of the course you will see how some more abstract mathematical techniques can be applied to the study of dynamics.

This course has strong links to the Differential Equations course from last term, and to the Vector Calculus course this term, and to almost every applied mathematics course next year.

Classical Dynamics (IIC)

This course expands on the Newtonian Dynamics course above, taking a more sophisticated mathematical approach and concentrating on the structure of the equations.

You will meet the Lagrangian formalism, an elegant approach which views dynamics as a variational problem, extremizing a quantity called the `action' over all possible paths of a system's evolution. The behaviour of mechanical systems near to equilbria will be analysed using linear algebra, and you will learn to analyse systems in terms of their normal modes. You will study Noether's Theorem, which relates conserved quantities in a system to symmetries of a system. A large part of the course will analyse the motion of rotating bodies in the Lagrangian framework, and finally you will learn about the Hamiltonian formalism, a beautiful approach which makes much of the underlying structure of mechanical systems clear, and provides a stunning connection between classical and quantum mechanics.

This course has obvious strong links to the Newtonian Dynamics course in IA, but also to Methods in IB, and Integrable Systems and Dynamical Systems in Part II. It provides an excellent introduction to many of the relativity and high energy physics courses in Part III.

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