Physics 106b
Topics in Classical Physics
Winter 2008

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Course description: The first half of Ph106b continues the study of classical mechanics begun in Ph106a. After a brief treatment of connections between classical and quantum physics, the course will be an introduction to nonlinear dynamical systems and chaos. Topics will include: chaotic maps in one dimension and two dimensions, Lyapunov exponents, canonical perturbation theory in Hamiltonian mechanics, invariant tori, the KAM theorem, hyperbolic and elliptic fixed points. The second half of Ph 106b will be the beginning of a course in electricity and magnetism, taught by Professor Eisenstein.

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Class meetings: Tuesdays and Thursdays 10:30-11:55 in 107 Downs, starting January 8.

John Preskill , 448 Lauritsen Laboratory, X-6691, email:

Teaching assistants:
Denis Bashkirov, email: , office hours: Tuesdays 6-8 pm, 269 Lauritsen.
Hee Joong Chung, email:
Haekong Kim, email:

Textbook:  Chaos in Dynamical Systems, by Edward Ott, 2nd edition. The material we will cover is mostly in Chapters 1, 2, 4,and 7. You may also want to refer to Chapter 11 of Classical Mechanics, by H. Goldstein, C. Poole, and J. Safko, 3rd edition. Lecture notes will be distributed for lectures that deviate substantially from the book.

Prerequisites: Ph106a or the equivalent (a course in Lagrangian and Hamiltonian mechanics).

Grading: Grades will be based on weekly problem sets, a midterm, and a final exam.

Homework: Homework will be posted here on Thursday, and will be due in class the following Thursday. If your homework will be late for a good reason, you may request an extension from the grader. Late homework will be accepted for half credit up until one week after the due date (no credit if your assignment is more than one week late).

Jan.  8: Canonical quantization, Hamiltonian-Jacobi theory as the classical approximation to the Schroedinger equation. Lecture notes.
Jan. 10: Path-integral formulation of quantum mechanics, the classical limit. Lecture notes.

Lecture notes for the following eight lectures are in four parts: Part 1, Part 2, Part 3, Part 4
Jan. 15: Sensitive dependence on initial conditions, Poincare section, tent map and shift map.
Jan. 17: Randomness, ergodicity, mixing, baker’s map.
Jan. 22: Cat map, chaotic billiards, logistic map, Lyapunov exponents.
Jan. 24: Integrable Hamiltonian dynamics and invariant tori.
Jan. 29: Canonical perturbation theory.
Jan. 31: KAM theorem, destruction of nearly rational tori, Henon-Heiles potential.
Feb.  5: Perturbations of the twist map, Poincare-Birkhoff theorem, elliptic, hyperbolic, and parabolic fixed points.
Feb.  7: Geometry of rational torus destruction: island chains, homoclinic tangle.

Homework assignments: 
Read lecture notes on the path integral formulation of quantum mechanics.
Problem Set 1, due 17 January 2008: Path integral (PDF). Solution: PDF
Read pages 1-30 of lecture notes; Ott Chapters 1 and 2.
Problem Set 2, due 24 January 2008: One-dimensional maps (PDF). Solution: PDF
Read lecture notes through page 42; Ott Sections 4.2, 4.3, 4.4, 6.1
Problem Set 3, due 31 January 2008: More on maps (PDF). Solution: PDF
Read lecture notes through page 79, Ott Chapter 7.
Problem Set 4, due 7 February 2008: Hamiltonian chaos (PDF). Solution: PDF

Midterm: available 8 February and due 14 February. Covers all material through Problem Set 4.
Cover page with midterm instructions: PDF. (You may open this now.)
Midterm exam: PDF. (Open only when you are ready to take the exam.)
Midterm solution: PDF

Midterm grades:
90-100             21
80-89               22
70-79               7
60-69               2
50-59               3
< 50                 1

56 exams, median=86