Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T17:19:46.222Z Has data issue: false hasContentIssue false
  • English
  • Français

On the equation grad f = M grad g

Published online by Cambridge University Press:  14 November 2011

Max Jodeit Jr  and
Peter J. Olver
Max Jodeit Jr
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Peter J. Olver
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The system of differential equations ∇f = M∇g, where M is a given square matrix, arises in many contexts. A complete solution to this problem in the case when M is a constant matrix is presented here. Applications to continuum mechanics and biHamiltonian systems are indicated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 116 , Issue 3-4 , 1990 , pp. 341 - 358
Copyright
Copyright © Royal Society of Edinburgh 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ball, J. M. and James, R.. Work in progress.Google Scholar
2Comtet, L.. Advanced Combinatorics; the Art of Finite and Infinite Expansions (Boston: D. Reidel, 1974).Google Scholar
3Dieudonné, J.. La dualité dans les espaces vectoriels topologiques. Ann. Sci. École Norm. Sup. (3) 59 (1942), 107–139.CrossRefGoogle Scholar
4Olver, P. J.. Conservation laws in elasticity. II. Linear homogeneous isotropic elastostatics. Arch. Rational Mech. Anal. 85 (1984), 131–160.CrossRefGoogle Scholar
5Olver, P. J.. Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107 (New York: Springer, 1986).CrossRefGoogle Scholar
6Olver, P. J.. Conservation laws in elasticity. III. Planar linear anisotropic elastostatics. Arch. Rational Mech. Anal. 102 (1988), 167–181.CrossRefGoogle Scholar
7Olver, P. J.. Canonical variables and integrability of biHamiltonian systems Phys. Lett. A (to appear).Google Scholar
8Schwartz, L.. Théorie des Distributions, nouvelle édition (Paris: Hermann, 1966).Google Scholar
9Turiel, F.-J.. Classification locale d'un couple de formes symplectiques Poisson-compatibles. Comptes Rendus Acad. Sci. Paris 308 (1989), 575–578.Google Scholar
10Turnbull, H. W. and Aitken, A. C., An Introduction to the Theory of Canonical Matrices (London; Blackie, 1932).Google Scholar
11Warner, F. W.. Foundations of Differentiate Manifolds and Lie Groups (Glenview, Ill.: Foresman, 1971).Google Scholar