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.*