Calculus of Variations and Tensor Calculus
Prerequisites: Linear algebra, e.g.
Math 5101; elementary
A physics course (e.g. Physics 133 or higher) would be helpful.
Variations and Tensor Calculus (Lecture
Notes) by U.H. Gerlach;
Calculus of Variations by I.M.Gelfand and Fomin;
Selected chapters from Gravitation by C.W. Misner, K.S. Thorne and J.A. Wheeler
Audience: Advanced undergraduates and graduates (Engineering, mathematics, physics)
Purpose: To develop the mathematical framework
surrounding dynamical systems,
including the mechanics of particles and of elastic and fluid media.
The development will focus on
(1) the important extremum principles in physics, engineering,
and mathematics and on
(2) the modern mathematical description for the kinematics
and dynamics of continuous media.
Instructor: Ulrich Gerlach: MW 124B/MW506; Telephone #: 292-5101 (dept.), 292-2560 (office), 292-7235 (office);
Description: I. Calculus of Variations (8 weeks):
problems in the calculus of variations.
Constraints and isoperimetric problems.
Variable end point problems.
Hamilton's principle, Lagrange's equations of motion.
Hamilton's equations of motion, phase space.
Action as the dynamical phase of a wave, the equation of Hamilton and Jacobi
Particle motion in the field of two attractive centers.
Helmholtz's equation in arbitrary curvilinear coordinates.
Rayleigh's quotient and the Rayleigh-Ritz method.
II. Tensor Calculus (6 weeks):
covectors and reciprocal vectors.
Tensors and tensor products.
Vector as a derivation.
Commutator of two vector fields.
Parallel transport of vectors on a manifold, the covariant differential.
Derivative of vectors and tensors
Strain-induced parallel transport in an elastic medium.
Strain as a deformation in the metric.
Parallel transport induced by a metric.
Tidal acceleration and the equation of geodesic deviation.