Mathematics 428/609D, Section 101

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

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.