Newton's conventional formulation of Classical Mechanics focuses attention on the forces acting on a system of particles and the second law of Newton then provides a way of calculating the subsequent position and motion of all the particles making up the system. This process can at times be rather awkward - particularly if there are constraints on the system. If we have, for example, a bead which is constrained to slide along a wire of known shape then the forces which constrain the bead to remain on the wire - the forces of constraint - are usually the reaction forces and they must be calculated using Newton's Laws for motion in say the x, y and z directions. Such a calculation may be awkward. However these constraint forces can be thought of as providing a purely geometrical constraint on the motion of the bead on the wire. If we can get away from having to work out the reaction forces, by using any convenient coordinate which incorporates the geometry of the problem (such as the distance moved along the wire) then we need never alculate the reaction (or constraint) forces. This elegant and beautiful reformulation of classical mechanics due to Lagrange and Hamilton, does exactly this and lies at the centre of the thinking about much modern physics. It allows us to choose any convenient set of coordinates to describe a problem and focuses on energies of a system- usually much easier to write down than the forces. We are thus are provided with a convenient and very practical way of analysing the motion of quite complicated systems. It also provides remarkable insights into the relations between the symmetries in a system and the conservation laws which hold. We can after all observe a conservation law in a scattering experiment and use this to deduce the symmetry of the underlying forces of nature. As a general rule in physics, a reformulation of any problem will usually offer new insights into its solution. We shall see also that the beautiful methods developed by Lagrange and Hamilton for Classical Mechanics are very close in spirit to the outlook of both Quantum Mechanics and Statistical Mechanics and Dirac's classic text on Quantum Mechanics draws heavily on the ideas which we shall develop in this module. Latterly the language and ideas of Lagrangian and Hamiltonian Mechanics have found fruit in the description of the behavior of certain Chaotic systems. The section of the module on the Calculus of Variations provides the mathematical background needed for a study of the General Theory of Relativity in year 4. |