Time:      MWF  12:40-1:35 pm

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: MW506; Telephone #:   292-5101 (dept.), 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.
                               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.
                               Tidal acceleration and the equation of geodesic deviation.