# 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