## Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Recitations: 1 session / week, 1 hour / session

## Prerequisites

You must complete *8.223 Classical Mechanics II* with a grade of C or better before taking this course. *8.03 Physics III: Vibrations and Waves* is also recommended as a prerequisite, although it is not required.

## Course Description

This course covers Lagrangian and Hamiltonian mechanics, systems with constraints, rigid body dynamics, vibrations, central forces, Hamilton-Jacobi theory, action-angle variables, perturbation theory, and continuous systems. It provides an introduction to ideal and viscous fluid mechanics, including turbulence, as well as an introduction to nonlinear dynamics, including chaos.

## Textbooks

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

Goldstein will serve as the main text for the majority of our material. However, we will also cover material that is not in Goldstein, in particular in our discussion of fluid dynamics and chaos. For the latter, there will be assigned readings from other sources.

For additional reading you may consider some of the following texts:

Landau, L. D., and E. M. Lifshits *Fluid Mechanics*. Butterworth-Heinemann, 1987. ISBN: 9780750627672. [Preview with Google Books]

Their first three chapters cover the material for our introduction to fluid mechanics. We will only use the third chapter for our discussion of viscosity.

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

Symon, Keith R. *Mechanics*. Addison-Wesley, 1971. ISBN: 9780201073928.

We will use a chapter from this book for reading material on non-viscous fluids.

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

An accessible yet detailed discussion of non-linear dynamics in both differential equations and iterated maps.

Landau, L. D., and E. M. Lifshits. *Mechanics*. Addison-Wesley, 1960.

Used for Classical Mechanics II and hence useful here for review.

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

A useful reference at a somewhat lower level than Goldstein, may be clearer for some topics.

## Problem Sets

Problem sets are an important part of this course. Sitting down and trying to reason your way through a problem not only helps you to learn the material deeply, but also develops analytical skills fundamental to a successful career in science. We recognize that students also learn a great deal from talking to and working with each other. We therefore encourage each student to make his / her own attempt on every problem and then, having done so, to discuss the problems with one another and collaborate on understanding them more fully. After any discussion, the solutions you write up and submit must reflect your own work.

## Exams

There will be a two–hour midterm and a three–hour final exam.

## Grading

Grades will be determined by a weighted average of:

ACTIVITIES | PERCENTAGES |
---|---|

Problem Sets | 35% |

Midterm Exam | 25% |

Final Exam | 40% |

The faculty may also take into account other qualitative measures of performance such as class participation, improvement, and effort.

## Subjects and Topics

SUBJECTS | TOPICS |
---|---|

Lagrangian and Hamiltonian Mechanics with Constraints | Euler-Lagrange Equations, Hamilton Equations, D'Alembert and Hamilton principles, Conservation Laws, holonomic and nonholonomic constraints, Lagrange multipliers |

Rigid Bodies & Rotations | Non-inertial coordinate systems, rotation matrices, Euler's theorem, Moment of Inertia Tensor, Euler Equations, Lagrangian for a spinning top with torque |

Vibrations and Oscillations | Simultaneous diagonalization of matrices for kinetic and potential energy. Using normal coordinates as generalized coordinates |

Canonical Transformations and Hamilton-Jacobi Equations | Generating functions for canonical transformations, invariants, Hamilton-Jacobi equation, action-angle variables. Kepler problem with action-angle variables. Three-Body problem and Lagrange points |

Perturbation Theory | Time dependent perturbation theory, periodic and secular changes, Adiabatic invariants |

Introduction to Fluid Mechanics | Dynamics for continuous systems. Hydrostatics, conservations laws, Euler equation, incompressible flows, and sound waves. Viscous flows and the Navier-Stokes equation. Reynolds number, Vortices, and Turbulence |

Introduction to Chaos and Nonlinear Dynamics | Fixed points, Bifurcation, and Limit cycles. Lorenz Equations. The Logistic Map. Fractals and Strange Attractors |