HAMILTONIAN MECHANICS

MAT461HS

Spring 2023

Time/location: TU 10-12/OI8180 and TH 11-12/BA1240
Instructor: Prof. Boris Khesin

Email: khesin@math.toronto.edu
Office: BA 6228
Office hours: WE 9-10:30 online

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, Hamilton-Jacobi theory, action-angle variables, and integrable systems.

Textbooks:

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

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

3) Notes on differential forms: Notes 1, Notes 2

4) The collection of V.Arnold's problems "Mathematical Trivium" 1991

Watch the Zoom talk "Closed problems session" on their solutions:
12:30pm, Tuesday March 28, 2023
https://utoronto.zoom.us/j/99576627828
Passcode: 448487


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. 7)

Problem Set 2 (due Tuesday, Feb. 28)

Problem Set 3 (due Thursday, Mar. 30)


List of project topics (due Tuesday, April 11)

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



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 Lagrangian and Hamiltonian settings of continuous systems. The hydrodynamical Euler equation.

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

PDF version of the course description