Calculus of Variations and Tensor Calculus
Credits: 3
Prerequisites: Linear algebra, e.g.
Math 5101; elementary
differential equations.
A
physics course (e.g. Physics 133 or higher) would be helpful.
Texts:
Calculus of
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);
e-mail: gerlach.1@osu.edu
Description: I. Calculus of Variations (8 weeks):
Classical
problems in the calculus of variations.
Euler's
equation.
Constraints
and isoperimetric problems.
Variable
end point problems.
Geodesics.
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):
Vectors,
covectors and reciprocal vectors.
Multilinear
algebra.
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.
Curvature.
Tidal
acceleration and the equation of geodesic deviation.