Course Details in 2025/26 Session

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Module Title LI Lagrangian and Hamiltonian Mechanics Physics and Astronomy Physics & Astronomy 03 00539 Nicola Wilkin Intermediate Level 10 Semester 2 LC Mathematics for Physicists 1A - (03 34459) LC Mathematics for Physicists 1B - (03 34462) none Lecture-24 hours Practical Classes and workshops-11 hours Guided independent study-65 hours Total: 100 hours 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. By the end of the module students should be able to: Identify the number of degrees of freedom in a mechanical system, select appropriate generalised coordinates and write down the kinetic and potential energies and a Lagrangian in terms of these generalised coordinates and generalised velocities; Identify the cyclic coordinates and all conserved quantities and write down the relevant conjugate canonical momenta and the Hamiltonian Integral and use these (together with Lagrange's equations) to solve mechanical problems; Analyse the central force problem and Kepler's gravitational problem using standard equations in polar coordinates; Determine the equilibrium configurations of a system with several degrees of freedom; Determine the frequencies and motions associated with the normal modes of small amplitude oscillations about equilibrium; Set up and solve for the extremals of straightforward problems in the Calculus of Variations and understand and use the links between the Calculus of Variations and Lagrangian Mechanics; Describe a system in terms of the generalised coordinates and the conjugate canonical momenta and write down the Hamiltonian of a system, starting from its Lagrangian; Solve Hamilton's equations in appropriate circumstances and use Conservation Laws; Construct a Canonical Transformation, calculate a Poisson Bracket and understand and the use of the Poisson Bracket in a Canonical Transformation; Write down the time evolution of a function of the dynamical variables in terms of the Poisson Brackets; Understand and solve the Hamilton Jacobi Equation in straightforward circumstances. 00539-01 : Exam : Exam (Centrally Timetabled) - Written Unseen (80%) 00539-02 : Assessed problems : Coursework (20%) Coursework (20%); 1.5 hour Examination (80%) none