This section includes the weekly problem sets and four larger projects. The projects are available online, and links are provided in the table below.
|1||Mechanics is more than equations of motion||8.1, 8.2 (from notation section)|
|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.
|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.
|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.
|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|
|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.
|25||Integrable systems||Assignment 9 (due in Ses #29): exercises 4.5, 4.6|
|27||Invariant curves, KAM theorem|
|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|
|32||Lie transforms and Lie series||Assignment 11 (due in Ses #36): exercises 5.26, 5.27, 5.30, 6.2|
|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|