MAT560: Physics for Mathematicians I (Fall 2007)

Instructor

Coordinates:

Outline

This is the first part of an ambitious two-part course in theoretical physics aimed at the graduate mathematics student. It is a fundamental part of the RTG Program in Geometry and Physics with the purpose of introducing many of the basic concepts, theories, and principles which form the basis of our current understanding of the Universe.

In the first semester we will cover

• Classical Mechanics:
  • Newtonian Mechanics
  • Lagrangian formalism and Action Principle
  • Symmetries
• Special relativity:
  • Spacetime diagrams
  • Physical peculiarities of Lorentz transformations
  • Covariant particle actions and background fields
• Electromagnetic theory:
  • Relativistic form
  • Electro- and Magneto-statics and dynamics
• Classical field theory:
  • Gauge Principle
  • Classical Yang-Mills theory

The course will be taught from a physical perspective (mathematical rigor is not emphasized) and is as such not a traditional mathematical physics course. The aim is to gain familiarity with and intuition for many of the concepts usually taught to a student of theoretical physics over the course of his/her undergraduate and graduate training.

The unorthodox nature of this program makes it impossible to follow a textbook. Nevertheless, I have assigned Theodore Frankel's "The Geometry of Physics: An Introduction" as a text. This book, as many others, is written to teach physicists mathematics and not the other way around. We will try to reverse-engineer and follow various sections of this book throughout the semester.

Due to the vast amount of material and the lack of centralized supporting literature, a lot of work will be required on the students' part to keep up. Besides the completion of regularly assigned homework sets, the student should read up on the topics being presented in the class.

Announcements

• No Class Monday, October 1st.
• No Class Wednesday, November 21st. (Correction day)

Assignments

1. Homework 1 due September 17 in class: Read chapters 1-3 and do problems 2.1(2), 2.5(2), 2.8(3), 2.10 (3,4), 3.1(3), 3.3(1,2).
2. Homework 2 due October 3 in class: Read chapters 9 and 10 and do all the problems in this (PDF) set.
3. Homework 3 due October 29 in class: Do the reading and all the problems in this (PDF) set.
4. Homework 4 (PDF) due November 26 in class.
5. Homework 5 (PDF) due December 19.

Resources

• In class I recommended taking a look at V.I. Arnold's "Mathematical Methods of Classical Mechanics. I will try to remember to put it on reserve in the library.
• Matt Young reminded me of the notes on Classical Field Theory and Supersymmetry by Daniel S. Freed in the 1996-1997 IAS program in Quantum Field Theory. As Matt points out, there is quite a bit of overlap with what we are/will be doing in class from a pure math perspective. (Thanks Matt!)
• Thanks go out again to Matt Young for pointing out this article on how to derive the Euler-Lagrange equation in a coordinate invariant manner.
• Some of the material on special relativity I am presenting in class is based on Bernard Schutz' book "A first course in general relativity" [Spires]