# Graduate Classical Mechanics

*Lecture Notes by Michael Fowler, UVa. *

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 make use of the more abstract mathematical formulation of dynamics, as developed in Arnold's book.

**Update!** Previously, we didn't discuss chaos, but now we've have added a chapter (22a) on the driven damped pendulum, which exhibits period doubling and chaos. We follow the order of presentation in Taylor's book, *Classical Mechanics*. We've also added a series of interactive applets, enabling the reader to explore the system over a wide parameter range. For appropriate values, the applets reproduce Taylor's figures.

## Lectures

1. An introductory statics lecture on the catenary

3.Fermat's Principle of Least Time

4. Hamilton's Principle and Noether's Theorem

8. A New Expression for the Action

11. Introduction to Liouville's Theorem

12. The Hamilton-Jacobi Equation

13. Adiabatic Invariants and Action-Angle Variables

19. One-Dimensional Crystal Dynamics

22. Resonant Nonlinear Oscillations

22a. Damped Driven Pendulum: Period Doubling and Chaos