Physical Mathematics
Phys 223c, Spring 2014,
“Physical Mathematics”
Instructors: Boris Shraiman 2317 Kohn Hall shraiman@kitp.ucsb.edu
and Madhav Mani rm 2117 Kohn Hall madhav.mani@gmail.com
Thursdays 4:00-6:30pm KITP (small seminar room)
Who needs to know how to solve equations when there is Mathematica and Matlab? As many have found out, computer graphics is great but still can’t substitute for understanding, and how does one know if the picture is right! This graduate course will aim to provide practical mathematical tools, which combine analytic approximations with numerical (Matlab) solutions for dealing with non-standard problems that are all too often encountered in doing original research.
- Approximate solutions to nasty equations, or the power of “Dominant balance”
- Polynomial eqns
- Nonlinear ODEs
(3 lectures)
- Approximating integrals
- “Easy”, “hard” and singular integrands
- Laplace method
(3 lectures)
- Boundary Layer theory and matched asymptotic expansions.
- Simple examples; Van Dyke’s matching rule
- Troubles with logarithms
- Examples from physics and engineering
(6 lectures)
- Multiple scale analysis
- Van der Pol relaxation oscillator
- Parametric instability (Mathieu eqn)
- Pattern forming instabilities: Amplitude and phase equations
- Perturbation theory “beyond all orders”.
(4 lectures)
- Elements of Dynamical Systems Theory
- Phase plane analysis
- Bifurcations
- Return maps
- Lyapunov exponents and chaos.
- Invariant measures, mixing and ergodicity
(4 lectures)
Texts: Michael Brenner, Harvard course notes
E.J. Hinch, “Perturbation methods”, Cambridge U . Press 1991
C.M. Bender and S.A. Orszag, “Advanced Mathematical Methods for Scientists and Engineers”, McGraw Hill, 1978
M. Cross and P.Hohenberg, Pattern formation outside of equilibrium
Rev. Mod. Phys, 65: 851–1112, 1993.
E. A. Jackson, “Perspectives of nonlinear dynamics”