Topics in Classical Physics

Winter 2008

**Go to home page for
Ph106a**

**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.

**Feedback:** If you
want to send a comment about the course, click here.

**Class meetings**:
Tuesdays and Thursdays 10:30-11:55 in 107

**Instructor:
**John Preskill
, 448 Lauritsen Laboratory, X-6691, email: preskill@theory.caltech.edu

Teaching assistants:

Hee Joong Chung, email: hjchung@caltech.edu

Haekong Kim, email: hkkim@caltech.edu

**Textbook:** *Chaos in Dynamical Systems*, by Edward Ott, 2^{nd} 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).

**Lectures:**

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

**Exams:**

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