Physics 106a
Topics in Classical Physics
Fall 2007

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Course description: A course in classical mechanics. Topics include calculus of variations and principle of least action, Lagrangian and Hamiltonian mechanics, central force motion, rigid bodies, small oscillations, canonical transformations, Hamilton-Jacobi theory, action-angle variables. (The first half of Ph 106b will be an introduction to chaotic behavior in classical dynamics.)

Class meetings: Tuesdays and Thursdays 10:30-11:55 in 107 Downs, starting October 2.

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

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:  Classical Mechanics, by H. Goldstein, C. Poole, and J. Safko, 3rd edition.

Prerequisites: A course in Newtonian mechanics. Familiarity with multivariable calculus and linear algebra.

Grading: Grades will be based on weekly problem sets (30%), a midterm (30%), and a final exam (40%).

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 (tentative schedule):
Oct.  2: Calculus of variations, brachistochrone problem.
Oct.  4: Constrained extremization, Lagrange multipliers.
Oct.  9: Principle of Least Action, the Lagrangian, holonomic constraints.
Oct. 11: Symmetries and conservation laws.
Oct. 16: Action of an electrically charged particle, the two-body problem.
Oct. 18: Kepler problem, differential cross section.
Oct. 23: Rutherford scattering, rigid body kinematics.
Oct. 25: Angular velocity, fictitious forces, inertia tensor.
Oct. 30: Euler equations, torque-free precession, stability of uniform rotation, Euler angles.
Nov.  1: Rigid body with one point fixed, the gyroscope.
Nov.  6: Sleeping top, coupled oscillators.
Nov.  8: Normal modes, linear molecule.
Nov. 13: Forced and damped oscillations.
Nov. 15: Hamilton’s equations, phase space, action as a function of coordinates.
Nov. 20: Maupertuis principle, adiabatic invariance of the action variable, virial theorem, Poisson bracket.
Nov. 27: Poisson’s theorem, Liouville’s theorem, canonical transformations.
Nov. 29: Generating functions, infinitesimal canonical transformations, symmetries and conservation laws as properties of flows in phase space.
Dec.   4: Canonical invariants, Hamilton-Jacobi theory.
Dec.   6: Integrable systems and action-angle variables.

Homework assignments: 
(Graded homework will be returned in class, and thereafter will be available for pickup outside 448 Lauritsen.)

Read Goldstein Chapters 1 and 2. (You may skim Sec. 1.4 through 1.6.)
Problem Set 1, due 11 October 2007: Calculus of variations (PDF). Solution: PDF
Problem Set 2, due 18 October 2007: Generalized coordinates and Lagrangian mechanics (PDF). Solution: PDF
Read Goldstein Chapter 3.
Problem Set 3, due 25 October 2007: Conservation laws and central force motion (PDF). Solution: PDF
Read Goldstein Chapter 4 and Chapter 5 through Sec. 5.6.
Problem Set 4, due 1 November 2007: Scattering, fictitious forces, and torque-free precession (PDF). Solution: PDF
Read Goldstein Chapter 5 from Sec. 5.7 and Chapter 6.
Problem Set 5, due 15 November 2007: Tops and small oscillations (PDF). Solution: PDF
Read Goldstein Chapter 8.
Problem Set 6, due 29 November 2007: Normal modes, etc. (PDF). (Note that the solution to Problem Set 5 is needed to do Problem Set 6.)
Read Goldstein Chapters 9, 10, and Sec. 12.5 (or Sec. 11.7 of the 2nd edition). Solution: PDF
Problem Set 7, due 6 December 2007: Hamiltonian stuff (PDF). Solution: PDF

Midterm: available 1 November and due 8 November. 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. Hee Joong graded problem 1 and Haekong graded problem 2. The mean was 76 and the standard deviation was 21.

Final: available 6 December and due 13 December. The final covers everything in the course, but emphasizes the material from after the midterm (including normal modes, canonical transformations and Hamilton-Jacobi theory).
Cover page with final exam instructions: PDF. (You may open this now.) Note that, in response to popular demand, you are permitted to consult Prof. Golwala’s on-line notes.
Final exam: PDF. (Open only when you are ready to take the exam.)
Final exam solution: PDF. Hee Joong graded problem1, Haekong graded problem 2,  Denis graded problem 3, and John graded problem 4. The median was 181/200. The exams are available for pickup outside 448 Lauritsen.

Grades for Ph106a were determined by the formula: 0.3*(HW) + 0.3*(Midterm) + 0.4*(Final), or 0.5*(HW) + 0.5*(Final), whichever was higher.
The ranges for letter grades were:

A+       95-100
A         90-94
A-        85-89
B+       80-84
B         70-79
B-        60-69
C+       50-59
C         40-49

There were 8 A+’s, 15 A’s, 17 A-‘s, 6 B+’s, 5 B’s, 2 B-‘s, 1 C (Total=54)