The Equivalence Problem and Canonical Forms for …olver/e_/ql.pdf · then form a 2kth order...
32
ADVANCES IN APPLIED MATHEMATICS 9, 226-257 (1988) The Equivalence Problem and Canonical Forms for Quadratic Lagrangians PETER J. OLVER* School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 The general equivalence and canonical form problems for quadratic variational problems under arbitrary linear changes of variable are formulated, and the role of classical invariant theory in their general solution is made clear. A complete solution to both problems for planar, first order quadratic variational problems is provided, including a complete list of canonical forms for the Lagrangians and corresponding Euler-Lagrange equations. Algorithmic procedures for determining the equivalence class and the explicit canonical form of a given Lagrangian are provided. Applica- tions to planar anisotropic elasticity are indicated. 0 1988 Academic PESS, IX. 1. INTR~DuCTI~N The basic problem of the calculus of variations is to determine the minima of a variational integral Here the integral is over a subdomain P of the space WP of independent variables x = (x,, . . . , xp), and the solutions u = f(x) are restricted to be in some appropriate space of vector-valued functions from II3 P to R 4, the space of dependent variables u = (ul,. . . , ~4). The Lagrungian L[u] is a function of x, u, and the derivatives of u, which, for simplicity, we take to be smooth in its arguments. The smooth minima of this variational problem are known to be solutions of the associated Euler-Lagrange equations E(L) = 0. Since the process of minimization does not depend on any particular coordinate system in use, it makes eminent sense to try to simplify the Lagrangian, and hence the associated Euler-Lagrange equations, as much *Research supported in part by NSF grant DMS 86-02004. 0196~8858/88 $7.50 Copyright 6 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. 226
![Rational Canonical Formbuzzard.ups.edu/...spring...canonical-form-present.pdfIntroductionk[x]-modulesMatrix Representation of Cyclic SubmodulesThe Decomposition TheoremRational Canonical](https://static.fdocuments.in/doc/165x107/6021fbf8c9c62f5c255e87f1/rational-canonical-introductionkx-modulesmatrix-representation-of-cyclic-submodulesthe.jpg)
