Hamiltonian Chaos

Sandro Wimberger, Parma University and INFN Section Milan Bicocca (Parma Group), Italy

Abstract:

The field of nonlinear dynamics and chaos has grown very much over the last decades and is becoming more and more relevant in different disciplines. We present a concise introduction to the field of nonlinear dynamics and chaos, suitable for graduate students of all branches of physics. We introduce the concepts of Hamiltonian dynamical systems' theory combining in a comprehensive way classical and quantum mechanical description. Motivations of the respective subjects and many examples from different fields in physics ease the understanding.

In the classical part, we review Hamilton-Jacobi theory as a fundamental basis for perturbational approaches (secular perturbation and KAM theory) and the definition of regularity and chaos (stable and unstable fixed points, Poincare-Birkhoff, Lyapunov exponents). The quantum part summerizes aspects of semiclassical theory (Wigner functions, Weyl symbols) as well as random matrix theory which is best suited to define chaos and complexity on a quantum level.

Literature:

V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer Verlag, New York, 1989)
Online book, P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, G. Vattay, Chaos: Classical and Quantum (Niels Bohr Institute, Copenhagen, 2012) at www.chaosbook.org
M.L. Mehta, Random matrices (Elsevier, Amsterdam, 2004)
S. Wimberger, Nonlinear Dynamics and Quantum Chaos: An Introduction (Springer, 2014)