Nonlinear dynamics ii: continuum systems

Lectures: 2 sessions / week, 1.5 hours / session

18.03 Differential Equations or 18.034 Honors Differential Equations; 8.02 Physics II: Electricity and Magnetism

This course introduces the basic ideas for understanding the dynamics of continuum systems, by studying specific examples from a range of different fields. Our goal will be to explain the general principles, and also to illustrate them via important physical effects. A parallel goal of this course is to give you an introduction to mathematical modeling.

The first part of the course will study diffusion, to demonstrate how continuum descriptions arise from averaging microscopic degrees of freedom. The equations of motion that we derive for continuum systems are typically nonlinear partial differential equations, for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray into dimensional analysis to see how it is possible to obtain qualitative information about a system without having to solve the full equations of motion.

We shall then study the 'calculus of variations', which is a minimization approach to finding solutions of continuous systems. We are familiar with these techniques for discrete systems and now adapt the ideas to help us with continuous systems. First, we examine the classical brachiostrome problem posed by Bernoulli, and then consider an array of problems in many different physical systems (e.g., shapes of soap films, bending of elastic beams, etc.).

In the second half of the course we will examine hydrodynamic problems. We will discuss derivations of Navier-Stokes-type equations and study particular solutions. In this context, we will introduce singular perturbation methods and survey hydrodynamic instabilities. In the last part, we will give an outlook on the mathematical description of solitons, active biological systems and topological defects.

Although there are no textbooks which follow the precise spirit of this course, there are literally hundreds of textbooks where the topics we will cover are discussed. For most lectures, typed notes are available. One book that will be useful frequently is:
Acheson, D. J. Elementary Fluid Dynamics. Oxford University Press, 1990. ISBN: 9780198596608. [Preview with Google Books]

Homework will be assigned roughly every two-three weeks. Each homework set will contain analytical and computational problems, and even the odd experiment. Assignments must be handed in by the start of class on the due date. The first unexcused late homework score will be multiplied by 75%. No subsequent unexcused late homework is accepted. You are welcome to discuss solution strategies and even solutions, but please write up the solution on your own. Be sure to support your answer by listing any relevant Theorems or by explaining important steps. Be as clear and concise as possible. I strongly encourage the computational problems to be written in MATLAB®.

There will be a mid-term exam, which will take place about the quarters of the way through the course. The exam will be a take-home, and will be in place of a homework set. There will be no final exam.

The ideas we will be discussing have applications to many fields, many of which we will not cover. To give you a chance to explore an area of interest to you, the course will require a final project, in which you explore in depth something of interest to you and within the course's scope. Final projects will be presented in class during the final two classes.

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