Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, however, of sheaf-theoretic nature. A new constructive method of finding a global Lagrangian for second-order ODEs on 2-manifolds is described on the basis of the solvability of the exactness equation for the Lepage 2-forms and the top-cohomology theorems. Examples from geometry and mechanics are discussed.
REFERENCES
1.
Anderson
, I. M.
and Duchamp
, T.
, “On the existence of global variational principles
,” Am. J. Math.
102
, 781
–867
(1980
).2.
Brajerčík
, J.
and Krupka
, D.
, “Variational principles for locally variational forms
,” J. Math. Phys.
46
, 052903
(2005
).3.
Dedecker
, P.
and Tulczyjew
, W. M.
, “Spectral sequences and the inverse problem of the calculus of variations
,” in Differential Geometry Methods and Mathematical Physics
, Lecture Notes in Mathematics Vol. 836, edited by García
, P. L.
, Pérez-Rendón
, A.
, and Souriau
, J. M.
(Springer
, Berlin
, 1980
), pp. 498
–503
.4.
Douglas
, J.
, “Solution of the inverse problem of the calculus of variations
,” Trans. Am. Math. Soc.
50
, 71
–128
(1941
).5.
Krupka
, D.
, “Lepagean forms in higher order variational theory
,” in Proceedings of the IUTAM-IAIMM Symposium on Modern Developments in Analytical Mechanics, Torino, Italy, 1982
, edited by Benenti
, S.
, Francaviglia
, M.
, and Lichnerowicz
, A.
(Academic Science Torino
, Torino
, 1983
), pp. 197
–238
.6.
Krupka
, D.
, “Variational sequences on finite order jet spaces
,” in Proceedings of the Conference on Differential Geometry and its Application, August 27–September 2, 1989
, edited by Krupka
, D.
and Švec
, A.
(World Scientific
, Brno, Czechoslovakia, Singapore
, 1990
), pp. 236
–254
.7.
Krupka
, D.
, “Variational sequences in mechanics
,” Calc. Var.
5
, 557
–583
(1997
).8.
Krupka
, D.
, Introduction to Global Variational Geometry
, Atlantis Studies in Variational Geometry Vol. 1 (Atlantis Press
, Amsterdam
, 2015
).9.
Krupka
, D.
, Krupková
, O.
, and Saunders
, D.
, “Cartan–Lepage forms in geometric mechanics
,” Int. J. Non-Linear Mech.
47
, 1154
–1160
(2012
).10.
Krupka
, D.
, Moreno
, G.
, Urban
, Z.
, and Volná
, J.
, “On a bicomplex induced by the variational sequence
,” Int. J. Geom. Methods Mod. Phys.
12
(5
), 1550057
(2015
).11.
Krupka
, D.
, Urban
, Z.
, and Volná
, J.
, “Variational submanifolds of Euclidean spaces
,” J. Math. Phys.
59
, 032903
(2018
).12.
Krupková
, O.
, “Lepagean 2-forms in higher order Hamiltonian mechanics, I. Regularity
,” Arch. Math. (Brno)
22
, 97
–120
(1986
).13.
Krupková
, O.
, The Geometry of Ordinary Variational Equations
, Lecture Notes in Mathematics Vol. 1678 (Springer
, Berlin
, 1997
).14.
Krupková
, O.
and Prince
, G. E.
, “Lepage forms, closed 2-forms and second-order ordinary differential equations
,” Russ. Math.
51
(12
), 1
–16
(2007
).15.
Krupková
, O.
and Prince
, G. E.
, “Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations
,” in Handbook of Global Analysis
, edited by Krupka
, D.
and Saunders
, D.
(Elsevier
, Amsterdam
, 2008
), pp. 837
–904
.16.
Lee
, J. M.
, Introduction to Smooth Manifolds
, Graduate Texts in Mathematics Vol. 218, 2nd ed. (Springer-Verlag
, New York
, 2002
); Errata at https://sites.math.washington.edu/∼lee/Books/ISM/.17.
Pommaret
, J. F.
, Partial Differential Equations and Group Theory
(Kluwer Academic Publishers
, Dordrecht
, 1994
).18.
Santilli
, R. M.
, Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics
(Springer-Verlag
, New York
, 1978
).19.
Takens
, F.
, “A global version of the inverse problem of the calculus of variations
,” J. Differ. Geom.
14
, 543
–562
(1979
).20.
Tonti
, E.
, “Variational formulation of nonlinear differential equations I
,” Acad. R. Belg. Bull. Cl. Sci.
55
, 137
–165
(1969
).21.
Tulczyjew
, W. M.
, “The Euler–Lagrange resolution
,” in Differential Geometry Methods and Mathematical Physics
, Lecture Notes in Mathematics Vol. 836, edited by García
, P. L.
, Pérez-Rendón
, A.
, and Souriau
, J. M.
(Springer
, Berlin
, 1980
), pp. 22
–48
.22.
Urban
, Z.
and Krupka
, D.
, “Foundations of higher-order variational theory on Grassmann fibrations
,” Int. J. Geom. Methods Mod. Phys.
11
(7
), 1460023
(2014
).23.
Urban
, Z.
and Volná
, J.
, “Exactness of Lepage 2-forms and globally variational differential equations
,” Int. J. Geom. Methods Mod. Phys.
16
, 1950106
(2019
).24.
Vinogradov
, A. M.
, “On the algebro-geometric foundations of Lagrangian field theory
,” Sov. Math. Dokl.
18
, 1200
–1204
(1977
).25.
Vinogradov
, A. M.
, “A spectral system associated with a non-linear differential equation, and the algebro-geometric foundations of Lagrangian field theory with constraints
,” Sov. Math. Dokl.
19
, 144
–148
(1978
).26.
Vitolo
, R.
, “Variational sequences
,” in Handbook of Global Analysis
, edited by Krupka
, D.
and Saunders
, D.
(Elsevier
, Amsterdam
, 2008
), pp. 1115
–1163
.© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
