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ISSN 1088-9485(e) ISSN 0273-0979(p)

Book Review

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Book Information

Author(s): Robert Bryant, Phillip Griffiths and Daniel Grossman
Title: Exterior differential systems and Euler-Lagrange partial differential equations
Additional book information: University of Chicago Press, 2003, 216 pp., (cloth) $45.00, ISBN 0-226-07793-4 ; (paper) $17.00, ISBN 0-226-07794-2

References:

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2.: Ball, J.M., Mizel, V.J., One-dimensional variational problem whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mech. Anal., 90 (1985) 325-388. MR 0801585 (86k:49002)
3.: Bryant, R.L., Chern, S.-S., Gardner, R.B., Goldschmidt, H.L., Griffiths, P.A., Exterior Differential Systems, Math. Sci. Res. Inst. Publ., vol. 18, Springer-Verlag, New York, 1991. MR 1083148 (92h:58007)
4.: Carathéodory, C., Über die Variationsrechnung bei mehrfachen Integralen, Acta Sci. Mat. (Szeged), 4 (1929) 193-216.
5.: Cartan, É., Sur la structure des groupes infinis de transformations, Oeuvres Complètes, part. II, vol. 2, Gauthier-Villars, Paris, 1953, pp. 571-714. MR 0753095 (85g:01032b)
6.: Gotay, M., An exterior differential systems approach to the Cartan form, Géométrie Symplectique et Physique Mathématique, P. Donato et al., eds., Birkhäuser, Boston, 1991, pp. 160-188. MR 1156539 (93e:58045)
7.: Griffiths, P.A., Exterior Differential Systems and the Calculus of Variations, Progress in Math. vol. 25, Birkhäuser, Boston, 1983. MR 0684663 (84h:58007)
8.: Hairer, E., Lubich, C., Wanner, G., Geometric Numerical Integration, Springer-Verlag, New York, 2002. MR 1904823 (2003f:65203)
9.: Jurás, M., Anderson, I.M., Generalized Laplace invariants and the method of Darboux, Duke Math. J., 89 (1997) 351-375. MR 1460626 (98h:58004)
10.: Kastrup, H.A., Canonical theories of Lagrangian dynamical systems in physics, Phys. Rep., 101 (1983) 1-167. MR 0733784 (85b:70020)
11.: Kosmann-Schwarzbach, Y., Les Théorèmes de Noether, Éditions de École Polytechnique, Palaiseau, France, 2004.
12.: Noether, E., Invariante Variationsprobleme, Nachr. Konig. Gesell. Wissen. Gottingen, Math.-Phys. Kl. (1918) 235-257. (See Transport Theory and Stat. Phys., 1 (1971) 186-207 for an English translation.) MR 0406752 (53:10538)
13.: Olver, P.J., Applications of Lie Groups to Differential Equations, Second Edition, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1993. MR 1240056 (94g:58260)
14.: Olver, P.J., Equivalence and the Cartan form, Acta Appl. Math., 31 (1993) 99-136. MR 1223167 (94i:58053)
15.: Olver, P.J., Moving frames -- in geometry, algebra, computer vision, and numerical analysis, in: Foundations of Computational Mathematics, R. DeVore, A. Iserles and E. Süli, eds., London Math. Soc. Lecture Note Series, vol. 284, Cambridge University Press, Cambridge, 2001, pp. 267-297. MR 1839146 (2002c:68091)
16.: Reichel, W., Uniqueness Theorems for Variational Problems by the Method of Transformation Groups, Lecture Notes in Mathematics, vol. 1841, Springer-Verlag, New York, 2004. MR 2068382
17.: Tsujishita, T., On variational bicomplexes associated to differential equations, Osaka J. Math., 19 (1982) 311-363. MR 0667492 (84b:58105)
18.: Vinogradov, A.M., The ${{\mathcal C}}$ -spectral sequence, Lagrangian formalism and conservation laws, I, II. J. Math. Anal. Appl., 100 (1984) 1-40, 41-129. MR 0739951 (85j:58150a), MR 0739952 (85j:58150b)
19.: Weyl, H., Geodesic fields in the calculus of variations for multiple integrals, Ann. Math., 36 (1935) 607-629. MR 1503239

Additional Information:

Reviewer(s):
Peter J. Olver
Affiliation: University of Minnesota

Review Information:
Journal: Bull. Amer. Math. Soc. 42 (2005), 407-412.

MSC (2000): Primary 35A30, 58A15, 58E30
PII: S 0273-0979(05)01062-1
Posted: April 1, 2005
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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