PHY622 · Mathematical Methods in Physics II

A knowledge wiki for PHY622 — Thammasat University graduate course, semester 2. Three arcs of beautiful mathematics: variational calculus, complex analysis, and group theory — each one a different answer to "what does physics optimize, integrate, or respect?"

The course in one paragraph

First arc: nature extremizes. Calculus of variations finds the curves that make functionals stationary via the Euler–Lagrange equation — geodesics, soap films, and all of mechanics through Hamilton's principle. Second arc: complex analyticity is a straitjacket so tight (Cauchy–Riemann) that a function's boundary values determine its interior (Cauchy's formula) and impossible real integrals fall to the residue theorem. Third arc: symmetry. Groups and their representations organize everything from crystal spectra to spin, culminating in Lie groups and the double cover SU(2) → SO(3) behind the electron's 720°.

Start here

• Lecture timeline — teaching order: variations → complex → groups
• Concept graph — the dependency map
• Conformal map explorer — watch the plane bend under analytic maps
• Brachistochrone racer — the cycloid wins, every time
• Contour integrator — residues evaluated numerically, arcs shown vanishing
• Worked examples — the catenary, summing series by residues, water's symmetry
• Quizzes · Glossary

Physics payoffs in this wiki family

Contour deformation is the heart of Landau damping; conformal maps solve potential flow around airfoils; the variational principle underlies every field theory you will meet.