Cohomology of the Lagrange complex. (English) Zbl 0654.58010
The concept of Lagrange complex was introduced by the author in the papers [C. R. Acad. Sci., Paris, Sér. A 280, 1295-1298 (1975; Zbl 0314.58018) and Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 24(1976), 1089-1096 (1977; Zbl 0352.58002)] in connection with the inverse problem of the calculus of variations of curves in a differential manifold. In the present paper the cohomology of the Lagrange complex is studied. By using the Ehresmann bundles of k-velocities together with an adapted form of theory of derivations of Frölicher and Nijenhuis the author gives a theorem relating the cohomology of the complex to de Rham cohomology of the manifold.
Reviewer: L.Maxim Răileanu
MSC:
| 58A99 | General theory of differentiable manifolds |
| 55N99 | Homology and cohomology theories in algebraic topology |
| 58J10 | Differential complexes |
| 58A12 | de Rham theory in global analysis |
Keywords:
Lagrange complex; inverse problem of the calculus of variations; cohomology; Frölicher; Nijenhuis; de Rham cohomologyReferences:
| [1] | LM. Anderson - T. Duchamp , On the existence of global variational principles , Amer. J. Math. , 102 ( 1980 ), pp. 781 - 868 . MR 590637 | Zbl 0454.58021 · Zbl 0454.58021 · doi:10.2307/2374195 |
| [2] | C. Ehresmann , Les prolongements d’ une variété différentiable , C.R. Acad. Sci. Paris , 233 ( 1951 ), pp. 598 - 600 . MR 44198 | Zbl 0043.17401 · Zbl 0043.17401 |
| [3] | A. Frölicher - A. Nijenhuis , Theory of vector valued differential forms , Nederl. Akad. Wetensch. Proc. Ser. A. , 59 ( 1956 ), pp. 338 - 359 . MR 82554 | Zbl 0079.37502 · Zbl 0079.37502 |
| [4] | G. Pidello - W.M. Tulczyjew , Derivations of differential forms on jet bundles , (to appear). MR 916711 | Zbl 0642.58004 · Zbl 0642.58004 · doi:10.1007/BF01762420 |
| [5] | F. Takens , A global version of the inverse problem of the calculus of variations , J. Differential Geometry , 14 ( 1979 ), pp. 543 - 562 . MR 600611 | Zbl 0463.58015 · Zbl 0463.58015 |
| [6] | W.M. Tulczyjew , Sur la différentielle de Lagrange , C.R. Acad. Sci. Paris Sér. A ., 280 ( 1975 ), pp. 1295 - 1298 . MR 377987 | Zbl 0314.58018 · Zbl 0314.58018 |
| [7] | W.M. Tulczyjew , The Lagrange differential , Bull. Acad. Polon. Sci. , 24 ( 1976 ). MR 650306 | Zbl 0352.58002 · Zbl 0352.58002 |
| [8] | W.M. Tulczyjew , The Lagrange complex , Bull. Soc. Math. France , 105 ( 1977 ), pp. 419 - 431 . Numdam | MR 494272 | Zbl 0408.58020 · Zbl 0408.58020 |
| [9] | A.M. Vinogradov , A spectral sequence associated with a nonlinear differential equation, and algebro-geometric foundation of Lagrangian field theory with constraints , Soviet Math. Dokl. , 19 ( 1978 ), pp. 144 - 148 . Zbl 0406.58015 · Zbl 0406.58015 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
