HAMILTONIAN MECHANICS

MAT461HS

Spring 2026

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, small oscillations, Hamiltonian formulation, canonical transformations, action-angle variables, integrable systems and an introduction to nonholonomic mechanics.

Textbooks:

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

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



Homework Assignments:
The full course mark consists of 3 assignments weighting together 50%, approximately 5% for in-class participation, and a final individual project weighting 45% (which consists of a survey minipaper 30% and a short discussion of its results with the instructor 15%). 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.

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 nonholonomic mechanics.

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

Also recommended: MAT367 (differential geometry)

PDF version of the course description