Graduate Classical Mechanics

Lecture Notes by Michael Fowler, UVa.

Fall 2015

Text:  Mechanics, Landau and Lifshitz.  

This is a very short book, but there's no padding.   The reviewer on the back of the newer edition (no doubt an expert in the field) states admiringly that "there is not a superfluous sentence" in this "masterpiece". In the present lectures, we provide fuller explanations to make the material (we hope) easier to follow on a first reading.

Summary

The course begins with an introduction to the Calculus of Variations, and Hamilton's Principle. Next we go to material covered in the last chapter of Landau, the Hamilton-Jacobi formalism that makes clear the intimate connection between classical mechanics and quantum mechanics. (The students are taking quantum simultaneously, so this works well in helping appreciation of classical mechanics, for example how least action is a limit of the sum over paths, and how classical adiabatic invariants are immediately understandable from a quantum perspective.)

The rest of the course follows the sequence of the book, beginning with Keplerian orbits, which we cover in more detail than Landau. (Perhaps his students were already familiar with this material?)

Then on to small oscillations, but including some interesting nonlinear systems, for example parametric resonance, and the ponderomotive force. Landau treats these analytically, using perturbation theory-type approximations. The last part of the course covers rotational motion: free body, tops, nutation, Coriolis, etc.

We have added some material using the direct Newtonian vectorial approach to Newtonian mechanics (as opposed to the Lagrangian formulation),following Milne. In discussing orbits, we derive Hamilton's equation, a very quick route to the Runge-Lenz vector. At the end of the course, we give Milne's elegant analysis of a ball rolling on a tilted rotating plane. The surprising cycloidal path can be derived in a few lines from Newton's equations. (It's tough to do this nonholonomic problem using Lagrangian methods.)

What don't we do?

We do not cover chaotic behavior, nor do we make use of the more abstract mathematical formulation of dynamics, as developed in Arnold's book.

Lectures

1. An introductory statics lecture on the catenary

2. The Calculus of Variations

3.Fermat's Principle of Least Time

4. Hamilton's Principle and Noether's Theorem

5. Mechanical Similarity

6. Hamilton's Equations

7. Poisson Brackets

8. A New Expression for the Action

9. Maupertuis' Principle

10. Canonical Transformations

11. Introduction to Liouville's Theorem

12. The Hamilton-Jacobi Equation

13. Adiabatic Invariants and Action-Angle Variables

14. Mathematics for Orbits

15. Keplerian Orbits

16. Elastic Scattering

17. Small Oscillations

18. Driven Oscillator

19. One-Dimensional Crystal Dynamics

20. Parametric Resonance

21. The Ponderomotive Force

22. Resonant Nonlinear Oscillations

23. Rigid Body Motion

24. Moments of Inertia

25. Rigid Body Moving Freely

26. Euler Angles

27. Eulers Equations

28 Non Inertial Frame, Coriolis effect

29. A Rolling Sphere on a Rotating Plane