Assignments

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