HAMILTONIAN MECHANICS

MAT461HS

Spring 2025

Time/location: TU 10-12 and TH 11-12
Instructor: Prof. Boris Khesin

Email: khesin@math.toronto.edu
Office: BA 6228
Office hours: TBA

Course description:
The course focuses on the key notions of classical mechanics: Newton equations, variational principles, Lagrangian formulation and Euler-Lagrange equations, the motion in a central force, the motion of a rigid body, Hamiltonian formulation, canonical transformations, action-angle variables, and integrable systems.

Textbooks:

1) Arnold: "Mathematical Methods of Classical Mechanics''

2) Goldstein, Poole, and Safko: "Classical Mechanics''



How to write mathematics by P.R. Halmos, Enseignement Math. vol.16(2) (1970), 123-152.


Lecture 6 on Similarity in physics


Homework Assignments:
There will be 3 assignments approximately weighting 20% each and a final individual project weighting 40% (which includes 2% of in-class participation), which together constitute the full course mark. No late assignments will be accepted.

Note: You must write your solutions yourself, in your own words. If your solution is aided by information from textbooks or online sources, you must properly quote these references.

Problem Set 1 (due Tuesday, Feb. 4)


Problem Set 2 (due Tuesday, Feb. 25)


Code of Behaviour / Plagiarism:
Students should become familiar with and are expected to adhere to the Code of Behaviour on Academic Matters.

Course Syllabus:
1. Newton equations. Lagrangian Mechanics. Energy and Momentum.
2. The two-body problem. Motion in a central field. The Kepler problem.
3. The Calculus of Variations. Euler-Lagrange equations.
4. Liouville's theorem, Poincare's recurrence.
5. Symmetries and Conservation laws. Noether's theorem.
6. Rigid Bodies. Euler's equations.
7. Hamiltonian Mechanics. Phase space. Symplectic manifolds. Poisson Brackets.
8. Lagrangian submanifolds. Integrable systems. Action-angle variables.
9. Billiard ball maps. Integrability of billiards in and the geodesic flows on quadrics.
10. Introduction to the hydrodynamical Euler equation.

Prerequisites:
MAT235Y1/MAT237Y1/MAT257Y1 (multivariable calculus),
MAT244H1/MAT267H1 (differential equations),
MAT223 (linear algebra)

PDF version of the course description