Introduction to the fundamental mathematical concepts and problems of classical dynamics.

What are Integrability, Chaos and Entropy? Why has time an arrow? The course is an introduction to classical dynamics and the mathematical problems it has lead to. We will discuss how simple deterministic systems can give rise to chaos and randomness. We will also discuss how chaotic systems with very large number of degrees of freedom can nevertheless give rise to simple statistical description and how dynamical equations which are symmetric with respect to reversing the direction of time can give rise to phenomena that are not any more time symmetric. These are fundamental problems of mathematical physics that were raised by Boltzmann, Poincare and others and which are still under intensive investigation.

The course is the first of introductory courses to mathematical physics. The second one deals with mathematical problems inspired by quantum physics and will be lectured on Spring 2021.

TARGET AUDIENCE

Mathematics students interested in the mathematics inspired by physics or physics students interested in the mathematical foundations of physics.

PREREQUISITES

Mathematics Students: Differential equations, Probability Theory, Measure and Integration are useful.

Physics students: Mathematical Methods for Physics 2.

No physics background is necessary but of course some is useful.

CONTENTS

1. Deterministic dynamics

2. Vector fields and flows

3. Examples: 1d motion, from simple to complex

4. Linearization

5. Invariant manifolds

6. Chaos: Arnold's cat, homoclinic tangle, Smale's solenoid

7. Symbolic dynamics

8. Ergodic theory, Lyapunov exponents

9. Integrable systems

10. Near integrable systems: KAM Theorem

11. Foundations of statistical mechanics, irreversibility and the arrow of time

COURSE MATERIAL

Lecture notes will be posted.