Classical Mechanics III

Readings

Goldstein = Goldstein, Herbert, Charles P. Poole, and John Safko. Classical Mechanics. Pearson, 2013. ISBN: 9781292026558.

Landau & Lifshitz = Landau, L. D., and E. M. Lifshits. Mechanics. Addison–Wesley, 1960.

Smits = Smits, Alexander J. A Physical Introduction to Fluid Mechanics. John Wiley And Sons Ltd, 1999.

Strogatz = Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, 1994. ISBN: 9780201543445.

Symon = Symon, Keith R. Mechanics. Addison–Wesley, 1971. ISBN: 9780201073928.

Thornton & Marion = Thornton, Stephen T., and Jerry B. Marion. Classical Dynamics of Particles and Systems. Cengage Learning, 2003. ISBN: 9780534408961.

WEEK #	READINGS
1 & 2	Lagrangian and Hamiltonian classical mechanics, symmetries and conservation laws. Use of generalized coordinates to handle constraints to motion on surfaces or curves. Read the following parts of Goldstein to review and extend concepts you have already seen in Classical Mechanics II: pp. 34–45, 55–56, 61–63, 334–42, then 343–47 and 353–56 (i.e., specific parts of sections 2.1–2.3, 2.6, 2.7, 8.1, 8.2, and 8.5.) If you are feeling rusty with Classical Mechanics II material you might want to review sections 1–10 and 40 of Landau & Lifshitz. On problem set 2 we will cover: D’Alembert’s principle and systems with holonomic and non–holonomic constraints. Use of Lagrange multipliers to determine forces of constraint. If you want to read ahead the corresponding reading is Goldstein: Sections 1.3–1.6, 2.4, 2.5, and 8.5.
3	Systems with Constraints: D’Alembert’s principle and systems with holonomic and non–holonomic constraints. Use of Lagrange multipliers to determine forces of constraint. The reading is Goldstein: Sections 1.3–1.6, 2.4, 2.5, and 8.5. Next up are Rigid Bodies which we’ll likely start at the end of this week and continue next week (they are not on problem set 2). The main readings for this entire topic are Goldstein chapter 4, sections 4.1, 4.2, 4.4, 4.6, and 4.9. Then read chapter 5, sections 5.1 and 5.3–5.7. The chapter 4 reading is on rigid body kinematics where many physics topics will be familiar to you. Goldstein emphasizes vector notation and discusses rotations as matrices. Chapter 5 is on rigid body dynamics. (Note that 4.3 is a linear algebra review. Other sections from chapter 4 and chapter 5 may be of interest, but the above are the most important ones.)
4	The main readings for Rigid Bodies are Goldstein chapter 4, sections 4.1, 4.2, 4.4, 4.6, and 4.9. (Note that 4.3 is a linear algebra review.) The chapter 4 reading is on rigid body kinematics where many physics topics will be familiar to you. Goldstein emphasizes vector notation and discusses rotations as matrices. Next read chapter 5, sections 5.1 and 5.3–5.7. Chapter 5 is on rigid body dynamics. We will not cover Poinsot’s construction, so you may skip this material on pp. 201–205 of section 5.6. (Other sections from chapter 4 and chapter 5 may be of interest, but the above are the most important ones.) If you find the reading in Goldstein too dense, you should consider reading Thornton & Marion chapter 11. I particularly recommend chapter 11, section 10 on the force–free motion of a symmetric top. Our next item to discuss will be principal axes for oscillating motion. If you would like to read ahead, the reading for this will be Goldstein chapter 6 sections 6.1–6.4.
5	The reading for Oscillations is Goldstein chapter 6, sections 6.1–6.4. We will spend a few weeks on our next subject: Canonical Transformations, the Hamilton–Jacobi equations, and Action–Angle Variables. The complete reading for this material is Goldstein chapter 9, sections 9.1–9.7 and then chapter 10, sections 10.1–10.6 and 10.8.
6	The reading for Canonical Transformations is Goldstein chapter 9, sections 9.1–9.7. (We will not discuss active infinitesimal canonical transformations with the same level of detail that Goldstein does in 9.6, but it is still good reading.) The reading on the Hamilton–Jacobi equations and Action–Angle Variables is Goldstein chapter 10 sections 10.1–10.6 and 10.8. We will cover more examples of this material on problem set 6.
7	The reading on Hamilton–Jacobi equations is Goldstein sections 10.1–10.5. The reading on Action–Angle Variables is Goldstein 10.6 and 10.8. You should also read section 10.7 pp. 457–60 (only up to Eq.10.109). After we finish discussing action angle variables our next subject will be Perturbation Theory, for which the reading is Goldstein chapter 12, sections 12.1–12.3.
8	For Perturbation Theory the reading assignment is Goldstein chapter 12, sections 12.1–12.3. Read Goldstein section 13.1 of chapter 13, on the transition from discrete to continuous systems. For Fluids, read sections 8.6–8.10, 8.13, and 8.14 from the Mechanics book by Symon.
9	For ideal fluids and sound waves, read sections 8.6–8.10, 8.13, and 8.14 from the Mechanics book by Symon. For fluids with viscosity the readings are from Landau and Lifshitz, Fluid Mechanics, Chapter II, sections 15–17 and 19–20. Further reading on fluids, including lots of worked examples and plenty of problems, can be found in the book “A physical introduction to Fluid Mechanics” by Alexander Smits. Michal Januszewski. “Flow Around a Sphere with (Lattice Boltzmann Fluid Simulation with Sailfish).” November 25, 2009. YouTube. Accessed April 16, 2015. https://www.youtube.com/watch?v=sv2ZA6SaORE Fluidmechanicsvideo. “Flow Past Cylinder - Science Experiment.” October 15, 2013. YouTube. Accessed April 16, 2015. https://www.youtube.com/watch?v=hrX11VtXXsU
10	We have finished our discussion of fluids, but there is one final problem on this in Problem Set 9. Further reading on fluids, including lots of worked examples and plenty of problems, can be found in the book “A physical introduction to Fluid Mechanics” by Alexander Smits. Read chapter 3 of Baker and Gollub on the Nonlinear Damped Forced Oscillator. Read the sections from chapter 3 of Strogatz on Bifurcations.
11	Read the sections from chapter 3 of Strogatz on Bifurcations. Read the sections from Strogatz on fixed points in two dimensions and limit cycles: 5.1–5.3, 6.1–6.5, 7.1–7.3, 8.1–8.2, and 8.4 (also have a look at 5.3 on love affairs, and 6.7 for pendulum phase space on a cylinder if you like). Read Goldstein section 11.8 on the logistic map and section 11.9 on fractals.