This section includes the weekly problem sets and four larger projects. The projects are available online, and links are provided in the table below.
SES # | TOPICS | ASSIGNMENTS |
---|---|---|
1 | Mechanics is more than equations of motion | 8.1, 8.2 (from notation section) |
Lagrangian mechanics | ||
2 | Principle of stationary action |
Assignment 1 (due in Ses #5): exercises 1.2, 1.3, 1.4, 1.5 Begin the Double Pendulum Project, exercise 1.39 (PDF), which will be due in Ses #13. |
3 | Lagrange equations | |
4 | Hamilton’s principle | |
5 | Coordinate transformations and rigid constraints |
Assignment 2 (due in Ses #8): exercises 1.8, 1.9 by hand, 1.11, 1.21 Remember to show all steps on 1.9 and 1.11. |
6 | Total-time derivatives and the Euler-Lagrange operator | |
7 | State and evolution: chaos | |
8 | Conserved quantities |
Assignment 3 (due in Ses #10): exercises 1.26, 1.27, 1.29 On 1.29, use the constraint formalism. Remember that the project on the double pendulum, exercise 1.39, is due in Ses #13. |
Rigid bodies | ||
9 | Kinematics of rigid bodies, moments of inertia | Assignment 4 (due in Ses #13): exercises 2.2, 2.3a, 2.3b, 2.4, 2.5, 2.6 |
10 | Generalized coordinates for rigid bodies | |
11 | Motion of a free rigid body |
Assignment 5 (due in Ses #16): exercises 2.11, 2.12, 2.13 Begin project on the rotation of Mercury, exercise 2.21, which will be due in Ses #21. |
12 | Axisymmetric top | |
13 | Spin-orbit coupling | |
Hamiltonian mechanics | ||
14 | Hamilton’s equations | |
15 | Legendre transformation, Hamiltonian actian |
Choose one of the following two projects. These projects are quite a bit of work, so the problem sets for the next few weeks are small. Exercise 3.14 (PDF): The Periodically-Driven Pendulum Exercise 3.15: Spin-orbit Surfaces of Section Your write-up of one of these is due in Ses #29. |
16 | Phase space reduction, Poisson brackets | |
17 | Evolution and surfaces of section | Assignment 6 (due in Ses #21): exercises 3.1, 3.3, 3.4 parts a and c, 3.5 |
18 | Autonomous systems: Henon and Heiles | |
19 | Exponential divergence, solar system | Assignment 7 (due in Ses #24): exercises 3.8, 3.10, 3.13 |
20 | Liouville theorem, Poincare recurrence | |
21 | Vector fields and form fields | |
22 | Poincare equations | |
Phase space structure | ||
23 | Linear stability |
Assignment 8 (due in Ses #27): exercises 4.1, 4.2, 4.3, 4.4 Also, remember that your project is due in Ses #29. |
24 | Homoclinic tangle | |
25 | Integrable systems | Assignment 9 (due in Ses #29): exercises 4.5, 4.6 |
26 | Poincare-Birkhoff theorem | |
27 | Invariant curves, KAM theorem | |
Canonical transformations | ||
28 | Canonical transformations, point transforms, symplectic conditions |
Last project, due in Ses #37: exercise 5.32 or 5.33, your choice. Note an error in 5.33: (1/2pi)*delta should be 2pi*delta in both places in the problem description. Assignment 10 (due in Ses #32): exercises 5.1, 5.4, 5.5, 5.14, 5.19 |
29 | Mixed-variable generating functions | |
30 | Time evolution is canonical | |
31 | Hamilton-Jacobi equation | |
32 | Lie transforms and Lie series | Assignment 11 (due in Ses #36): exercises 5.26, 5.27, 5.30, 6.2 |
Perturbation theory | ||
33 | Perturbation theory with Lie series | |
34 | Small denominators and secular terms, pendulum to higher order and many degrees of freedom | |
35 | Nonlinear resonances, reading the Hamiltonian, resonance overlap | |
36 | Second-order resonances, stability of the vertical equilibrium | |
37 | Adiabatic invariance and adiabatic chaos |