| Unit name | Mathematical Methods 404 |
|---|---|
| Unit code | PHYSM0400 |
| Credit points | 10 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 1B (weeks 7 - 12) |
| Unit director | Professor. Dave Newbold |
| Open unit status | Not open |
| Pre-requisites |
PHYS33010 Mathematical Methods 301. |
| Co-requisites |
None |
| School/department | School of Physics |
| Faculty | Faculty of Science |
Calculus of variations: some typical variational problems (VP). Concept of functional. The simplest variational problem. Euler-Lagrange equation. Free end-points and natural boundary conditions. Extension to VPs with higher derivatives and to VPs with several dependent variables. Hamilton's principle in mechanics. Noethers Theorem relating symmetries and conservation laws. Extension to VPs with several independent variables. Hamilton's principle for fields. Example: waves on a string. VPs with subsidiary conditions and Lagrange multiplier method. Asymptotics: concept of asymptotic expansion (AE). Numerical illustrations. Poincare's definition of AE. Integration-by-parts method and Laplace method for AE of functions defined by integrals. Laplace integrals and Watson's Lemma. Stationary phase approximation for oscillatory integrands. Application to dispersive waves; group velocity. Method of steepest descents (if time permits). Green functions: Green functions for second-order ordinary differential equations on a finite domain a<_x<_b. Bilinear expansion of Green function. Contour integral representation in case of infinite domain. Green function for partial differential equations: Poisson equation, Helmholtz equation. Retarded Green function for wave equation. Contour integral representation incorporating causal and outgoing-wave properties. Propagator for time-dependent Schrodinger equation; its perturbative expansion leading to non-relativistic Feynman diagrams (if time permits).
