# Mathematical Classical Mechanics

### Department of Mathematics,  University of British Columbia

Prerequisites: This course is intended to complement classical mechanics courses like Physics 206 in the sense that the physical background will be developed but the emphasis will be on the resulting mathematical analysis. Students should already have some experience with rigorous mathematics (like Math 320 and 321) and with classical mechanics (like Physics 206) although these prerequisites may be waived at the discretion of the instructor.

## Instructor

### Joel Feldman

 E-mail feldman@math.ubc.ca Office Math 221 Phone 822-5660 Home page http://www.math.ubc.ca/~feldman/ Office hours Mon 10:00-11:00, Wed 10:00-11:00, Fri 10:00-11:00

## Text

• V. I. Arnold, Mathematical Methods of Classical Mechanics,

I will post all handouts, problem sets, etc. on the web here.

## Other References

• G. Gallavotti, The Elements of Mechanics.
• H. Goldstein, Classical Mechanics.

## Topics

1. Newtonian Mechanics:
The principles of relativity and determinacy, the gallilean group, Newton's equations
Examples: the harmonic oscillator, pendulum and central fields
An introduction to phase space, conservation of energy, momentum and angular momentum
2. Constraint Free Lagrangian Mechanics:
Variational problems and the Euler-Lagrange equation
The lagrangian and Hamilton's principle of least action
The hamiltonian and Hamilton's equations
Liouville's theorem
Poincaré recurrence theorem
3. Lagrangian Mechanics on Manifolds:
The introduction of manifolds through constraints
Differentiable manifolds and tangent bundles
Lagrangian dynamics
Symmetry and Conservation laws: Noether's theorem
4. Differential Forms:
Exterior algebra, differential forms on manifolds, exterior differentiation, vector analysis
Chains, integration of differential forms
Stokes' theorem
Poincaré lemma

## Grading

• There will be weekly problem sets accounting for about 50% of the final mark.
• The final exam will account for about 50% of the final mark.
• Grades will probably be scaled.

## Policies

• The final examination will be strictly closed book: no formula sheets or calculators will be allowed.
• There is no supplemental examination in this course.
• Late homework assignments normally receive a grade of 0. Missing a homework normally results in a mark of 0. Exceptions may be granted in two cases: prior consent of the instructor or a medical emergency.