Firstly, what is classical mechanics? Classical mechanics is that part of physics that describes the motion of large-scale bodies (much larger than the Planck length) moving slowly (much slower than the speed of light).The paradigm example is the motion of the celestial bodies; this is not only the oldest preoccupation of science, but also has very important practical applications which continue to the present day. We shall have much to say about this example in this class.

More than this, classical mechanics is a complete mathematical theory of the universe as it appears at such scales, with a rich mathematical structure; it has been studied by almost every great mathematician from Newton to Poincaré. Precisely predicting the motion of the celestial bodies was the major scientific challenge of their time: making maps and calendars, safely navigating the seas, exploration and trade, predicting the tides, all depended on such data. In a world before accurate timekeeping, the heavens were the only means of measuring the passing of time. Such was its importance that the British parliament offered a prize of £3 million (in today’s money) for a sufficiently accurate solution. So much of modern mathematics had its genesis in attempts to solve this problem.

Moreover, a deeper understanding of the structure of classical mechanics reveals unexpected analogies that clarify the otherwise mystifying formalism of quantum mechanics: we shall see the classical analogs of anticommuting variables, the Schrödinger equation, and even the uncertainty principle! Similarly, one can see how even Newtonian gravity can be formulated in terms of ‘bending’ of space and time. This course
will study these structures in a modern way, in the language of symplectic manifolds with special emphasis on geometry and symmetry.

We shall mostly be referring to the beautiful book Mathematical Methods of Classical Mechanics by V. Arnol’d. While excellent for giving physical motivation, it is sadly very out of date in many places, and the ideas can be greatly clarified using modern terminology and notation. Be sure you have the second edition, since the most interesting material is lumped in the appendices at the end. We shall also be making reference to the notes by da Silva: all further administrative information may be found on the web page. The level of rigor will be somewhere between mathematics and physics: we will state definitions and theorems, but we won’t necessarily give complete proofs of all of them. We’ll leave some of the hypotheses of the theorems implicit, for instance, all functions will be assumed infinitely differentiable unless otherwise stated.

(1) 학부 고전역학
(2) 학부 수리물리
(3) 학부 양자역학

중간시험: 20%
기말시험: 30%
숙제: 30%
출석: 20%
현재로서는 변동 가능함.

1. Mathematical Methods of Classical Mechanics, Vladimir I. Arnold.
2. CLASSICAL MECHANICS AND SYMPLECTIC GEOMETRY, Maxim Jeffs.
3. Classical Mechanics and Geometry, Si Li.
4. Introduction to symplectic mechanics, Zaidni Azeddine.
5. Classical Dynamics (University of Cambridge Part II Mathematical Tripos), D. Tong.
6. CLASSICAL MECHANICS, Herbert Goldstein, Charles P. Poole, and John L. Safko.

1. Lagrangian formulation & Noether's theorem
2. Hamiltonian formulation & symmetries
3. Canonical transformations
4. Action-angle variables and adiabatic invariants
5. Hamilton-Jacobi equation

6. Vector field and Differential form
7. Cartan formula and Poincaré lemma
8. Symplectic form
9. Geometry of canonical transformations
10. Symplectic Manifold
11. Moment Map

https://www.youtube.com/@tobiasjosborne/playlists

Tobias Osborne 교수의 Symplectic geometry & classical mechanics 관련 Youtube 동영상 참조.