×

zbMATH — the first resource for mathematics

Equivalence and the Cartan form. (English) Zbl 0794.49041
Summary: An investigation into the Cartan form and nondegeneracy conditions for field-theoretic Lagrangians based on the Cartan equivalence method is presented.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
70H99 Hamiltonian and Lagrangian mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, I.M., Aspects of the inverse problem of the calculus of variations,Arch. Math. (Brno)24 (1988), pp. 181-202. · Zbl 0674.58017
[2] Anderson, I.M.,The Variational Bicomplex, Academic Press, to appear. · Zbl 0772.58013
[3] Anderson, I.M., Kamran, N., and Olver, P.J., Internal, external and generalized symmetries,Adv. in Math. (to appear). · Zbl 0809.58044
[4] B?cklund, A.V., Ueber Flachentransformationen,Math. Ann. 9 (1876), pp. 297-320. · JFM 07.0233.02
[5] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity,Arch. Rational Mech. Anal. 63 (1977), pp. 337-403. · Zbl 0368.73040
[6] Ball, J.M., Currie, J.C., and Olver, P.J., Null Lagrangians, weak continuity, and variational problems of arbitrary order,J. Func. Anal. 41 (1981), pp. 135-174. · Zbl 0459.35020
[7] Betounes, D.E., Extension of the classical Cartan form,Phys. Rev. D 29 (1984), pp. 599-606.
[8] Bodnar, A.C., The Cartan equivalence problem for first order Lagrangians inm independent andN dependent variables, MSc Thesis, Univ. of Waterloo, Ontario, Canada, 1986.
[9] Bryant, R.L., On notions of equivalence of variational problems with one independent variable,Contemp. Math. 68 (1987), pp. 65-76. · Zbl 0641.49001
[10] Carath?odory, C., ?ber die Variationsrechnung bei mehrfachen Integralen,Acta Sci. Mat. (Szeged)4 (1929), pp. 193-216. · JFM 55.0900.01
[11] Cartan, E., Les probl?mes d’?quivalence, inOeuvres Compl?tes, part. II, vol. 2, Gauthiers-Villars, Paris, 1952, pp. 1311-1334.
[12] Cartan, E., Sur un probl?me d’?quivalence et la th?orie des espaces m?triques g?n?ralis?s, inOeuvres Compl?tes, part, III, vol. 2, Gauthiers-Villars, Paris, 1955, pp. 1131-1153.
[13] Dedecker, P., Calcul des variations et topologie alg?brique,M?m. Soc. Roy. Sci. Li?ge 19 (1957), pp. 1-216.
[14] De Donder, T.,Th?orie Invariantive du Calcul de Variations, Gauthier-Villars, Paris, 1935.
[15] Egecioglu, ?. and Remmel, J.B., Symmetric and antisymmetric outer plethysms of Schur functions,Atomic Data and Nuclear Data Tables 32 (1985), pp. 157-196.
[16] Federer, H.Geometric Measure Theory, Springer-Verlag, New York, 1969. · Zbl 0176.00801
[17] Frame, J.S., Robinson, G. de B., and Thrall, R.M., The hook graphs of Sn,Canad. J. Math. 6 (1954), pp. 316-324. · Zbl 0055.25404
[18] Gardner, R.B.,The Method of Equivalence and Its Applications, SIAM, Philadelphia, 1989. · Zbl 0694.53027
[19] Gardner, R.B. and Shadwick, W.F., Equivalence of one dimensional Lagrangian field theories in the plane I, in D. Ferus et al. (eds.),Global Differential Geometry and Global Analysis, Lecture Notes in Math. 1156, Springer-Verlag, New York, 1985, pp. 154-179. · Zbl 0582.58031
[20] Gel’fand, I.M. and Fomin, S.V.,Calculus of Variations Prentice-Hall, Englewood Cliffs, NJ, 1963.
[21] Gotay, M., A multisymplectic framework for classical field theory and the calculus of variations: I. Covariant Hamiltonian formalism, in M. Francaviglia (ed.), Mechanics, Analysis and Geometry: 200 Years after Lagrange, Elsevier, New York, 1991, pp. 203-235. · Zbl 0741.70012
[22] Gotay, M., An exterior differential systems approach to the Cartan form, in P. Donato etal. (eds.), G?om?trie Symplectique et Physique Math?matique, Birkh?user, Boston, 1991.
[23] Griffiths, P.A.,Exterior Differential Systems and the Calculus of Variations, Progress in Math. Vol. 25, Birkh?user, Boston, 1983. · Zbl 0512.49003
[24] Hadamard, J., Sur une question de calcul des variations,Bull. Soc. Math. France 30 (1902), pp. 253-262. · JFM 33.0387.02
[25] Hsu, L., Kamran, N. and Olver, P.J., Equivalence of higher order Lagrangians II. The Cartan form for particle Lagrangians,J. Math. Phys. 30 (1989), pp. 902-906.
[26] Ibragimov, N.H.,Transformation Groups Applied to Mathematical Physics, D. Reidel, Dordrecht, 1985. · Zbl 0558.53040
[27] Kamran, N., Contributions to the study of the equivalence problem of ?lie Cartan and its applications to partial and ordinary differential equations,M?m. Cl. Sci. Acad. Roy. Belg. 45 (1989), Fac. 7. · Zbl 0721.58001
[28] Kamran, N., and Olver, P.J., Equivalence problems for first order Lagrangians on the line,J. Differential Equations 80 (1989), pp. 32-78. · Zbl 0677.49034
[29] Kamran, N., and Olver, P.J., Le probl?me d’?quivalence ? une divergence pr?s dans le calcul des variations des int?grales multiples,C. R. Acad. Sci. Paris S?rie I Math. 308 (1989), pp. 249-252. · Zbl 0674.49029
[30] Kamran, N. and Olver, P.J., Equivalence of higher order Lagrangians I. Formulation and reduction,J. Math. Pures Appl. 70 (1991), pp. 369-391. · Zbl 0746.58012
[31] Kamran, N. and Olver, P.J., Equivalence of higher order Lagrangians in. New invariant differential equations,Nonlinearity 5 (1992), pp. 601-621. · Zbl 0783.58014
[32] Kastrup, H.A., Canonical theories of Lagrangian dynamical systems in physics,Phys. Rep. 101 (1983), pp. 1-167.
[33] Kl?tzler, R.,Mehrdimensionale Variationsrechnung, Birkh?user Verlag, Basel, 1970.
[34] Krupka, D., Lepagean forms in higher order variational theory, in S. Benenti, M. Francaviglia, and A. Lichnerowicz (eds.),Proceedings of IUTAM-ISIMM Symposium on Modern Developments in Analytic Mechanics, Acta Acad. Sci. Taurinensis 117 (Suppl.) (1983), pp. 197-238. · Zbl 0572.58003
[35] Krupka, D. and ?t?p?nkov?, O., On the Hamilton form in second order calculus of variations, in M. Modugno (ed.)Proceedings of the Meeting ?Geometry and Physics?, Pitagora Editrice, Bologna, 1983, pp. 85-101.
[36] Littlewood, D.E.,The Theory of Group Characters, Oxford Univ. Press, New York, 1958. · Zbl 0079.03604
[37] Morrey, C.B., Jr.,Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. · Zbl 0142.38701
[38] Olver, P.J., Conservation laws and null divergences,Math. Proc. Cambridge Philos. Soc. 94 (1983), pp. 529-540. · Zbl 0556.35021
[39] Olver, P.J.,Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. · Zbl 0588.22001
[40] Olver, P.J., The equivalence problem and canonical forms for quadratic Lagrangians,Adv. Appl. Math. 9 (1988), pp. 226-257. · Zbl 0654.49002
[41] Olver, P.J., Canonical elastic moduli,J. Elasticity 19 (1988), pp. 189-212. · Zbl 0658.73014
[42] Olver, P.J., Canonical anisotropic elastic moduli, in J.J. Wu, T.C.T. Ting, and D.M. Barnett (eds.)Modern Theory of Anisotropic Elasticity and Applications, SIAM, Philadelphia, 1991, pp. 325-339.
[43] Rund, H.,The Hamilton-Jacobi Theory in the Calculus of Variations, Van Nostrand, Princeton, NJ, 1966. · Zbl 0141.10602
[44] Shadwick, W.F., The Hamiltonian formulation of regular rth order Lagrangian field theories,Lett. Math. Phys. 6 (1982), pp. 409-416. · Zbl 0514.58013
[45] Sivaloganathan, J., The generalised Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity,Arch. Rational Mech. Anal. 88 (1988), pp. 347-367. · Zbl 0709.73014
[46] Terng, C.L., Natural vector bundles and natural differential operators,Amer. J. Math. 100 (1978), pp. 775-828. · Zbl 0422.58001
[47] Terpstra, F.J., Die Darstellung biquadratische Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung,Math. Ann. 116 (1939), pp. 166-180. · Zbl 0019.35203
[48] Turnbull, H.W., Gordan’s theorem for double binary forms,Proc. Edinburgh Math. Soc. 41 (1922/23), pp. 116-127.
[49] Turnbull, H.W., A geometrical interpretation of the complete system of the double binary (2,2) form V,Proc. Roy. Soc. Edinburgh 44 (1923/24), pp. 23-50. · JFM 49.0687.01
[50] Weyl, H., Geodesic fields in the calculus of variations for multiple integrals,Ann. Math. 36 (1935), pp. 607-629. · Zbl 0013.12002
[51] Weyl, H.,Classical Groups, Princeton Univ. Press, Princeton, NJ, 1946. · Zbl 1024.20502
[52] Krupka, D., Some geometric aspects of variational problems in fibered manifolds,Fac. Sci. Nat. Univ. Purkyniane Brunensis 14 (1973), #10.
[53] Krupkov?, O., Lepagean 2-forms in higher order Hamiltonian mechanics I. Regularity,Arch. Math. Brno 22 (1986), pp. 97-120. · Zbl 0637.58002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.