Mathematical Methods of Science, Fall 2014

Important Announcements  The final exam is scheduled on Wed, Dec 24, at 10:30 a.m.,  in room 1001

Instructor: Vladlen Timorin
Office: 1007
Office hours: 3:30-4:50 pm (or by appointment)
e-mail: vtimorin --at-- hotmail.com

Textbooks:
V.I. Arnold, Mathematical Methods of Classical Mechanics
V.I. Arnold, Lectures on Partial Differential Equations
(copies of selected chapters will be provided upon request)

Lecture notes (check for updates!)

 Dec 12, 2014 (PDF, 394 Kb)

 

 

Syllabus

We discuss mathematical ideas coming from Science, notably from Physics. Rather than focusing on applications of Mathematics in Science, we focus on applications of ideas from Physics and "real life" to Mathematics. Our major objective for this course is to understand the motivations behind various notions of Fundamental Mathematics. The course will cover a selection from the following topics (the list provided below is about twice bigger than what fits into for a semester course):

• Geometric optics: the least time principle of Fermat, the Huygens principle, the fourth Hilbert problem
• Action functions and Hamiltonians: the least action principle, the Legendre transform, Hamiltonians vs. Lagrangians, the Hamilton-Jacobi equations
• Foundations of symplectic geometry: reminder on manifolds, vector fields, differential forms; symplectic structures, the Darboux straightening theorem, the Poincare recurrence theorem
• Canonical formalism: generating functions, the Hamilton-Jacobi method of integration, the first-order PDEs vs. Hamiltonian systems
• Integrable systems: Poisson structures and first integrals, integrability, the Liouville-Arnold theorem, Lie-Poisson structures, Lax pairs
• Partial differential equations of the first order: quasilinear equations, envelopes, the Monge cones, the Cauchy-Kovalevskaya theorem
• Some second order PDEs: the analytic classification, the wave equation, the d'Alembert and Fourier methods, the superposition principle, iterative methods, derivations for some equations of Mathematical Physics, classical vector calculus
• The Laplace and Poisson equations: harmonic functions, electrostatics, the Mean Value theorem, the Poisson formula, the Harnack inequalities, the Poincare-Perron method
• Distributions and fundamental solutions: spaces of distributions, the Fourier transform, fundamental solutions of linear differential operators with constant coefficients, convolutions, single and double layer potentials
• The eigenvalue problems for linear differential operators: eigenvalues of the Laplacians, spherical harmonics, the Sturm-Liouville problem, reduction to integral equations