The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulation for a subsystem of the given PDE. No constraints on the differential order or number of dependent or independent variables are assumed. The proof follows a recent observation of Bridges, Hydon, and Lawson [Math. Proc. Cambridge Philos. Soc. 148(01), 159–178 (2010)] and generalizes an older result of Henneaux [Ann. Phys. 140(1), 45–64 (1982)] from ordinary differential equations (ODEs) to PDEs. Uniqueness of the variational formulation is also discussed.
REFERENCES
1.
I.
Anderson
and G.
Thompson
, “The inverse problem of the calculus of variations for ordinary differential equations
,” Memoirs of the American Mathematical Society: Number 473
(AMS
, Providence, RI
, 1992
).2.
I. M.
Anderson
, “Aspects of the inverse problem to the calculus of variations
,” Arch. Math. (Brno)
24
(4
), 181
–202
(1988
).3.
I. M.
Anderson
, The Variational Bicomplex
(1989
).4.
I. M.
Anderson
, “Introduction to the variational bicomplex
,” in Mathematical Aspects of Classical Field Theory
, edited by M. J.
Gotay
, J. E.
Marsden
, and V.
Moncrief
, Contemporary Mathematics
Vol. 132
(American Mathematical Society
, Providence, RI
, 1992
), pp. 51
–73
.5.
I. M.
Anderson
and T. E.
Duchamp
, “Variational principles for second-order quasi-linear scalar equations
,” J. Differ. Equations
51
(1
), 1
–47
(1984
).6.
A.
Ashtekar
, L.
Bombelli
, and O.
Reula
, “The covariant phase space of asymptotically flat gravitational fields
,” in Mechanics, Analysis and Geometry: 200 Years After Lagrange
North-Holland Delta Series
, edited by M.
Francaviglia
and D.
Holm
(North-Holland
, Amsterdam
, 1991
), pp. 417
–450
.7.
G.
Barnich
, F.
Brandt
, and M.
Henneaux
, “Local BRST cohomology in gauge theories
,” Phys. Rep.
338
(5
), 439
–569
(2000
).8.
G.
Barnich
, M.
Henneaux
, and C.
Schomblond
, “Covariant description of the canonical formalism
,” Phys. Rev. D
44
(4
), R939
–R941
(1991
).9.
T. J.
Bridges
, P. E.
Hydon
, and J. K.
Lawson
, “Multisymplectic structures and the variational bicomplex
,” Math. Proc. Cambridge Philos. Soc.
148
(01
), 159
–178
(2010
).10.
R. L.
Bryant
and P. A.
Griffiths
, “Characteristic cohomology of differential systems. I. General theory
,” J. Am. Math. Soc.
8
(3
), 269
–307
(1995
).11.
Č.
Crnković
and E.
Witten
, “Covariant description of canonical formalism in geometrical theories
,” in Three Hundred Years of Gravitation
, edited by S. W.
Hawking
and W.
Israel
(Cambridge University Press
, Cambridge
, 1987
), pp. 676
–684
.12.
J.
Douglas
, “Solution of the inverse problem of the calculus of variations
,” Trans. Am. Math. Soc.
50
, 71
–128
(1941
).13.
G.
Giachetta
, L.
Mangiarotti
, and G.
Sardanashvily
, “Cohomology of the infinite-order jet space and the inverse problem
,” J. Math. Phys.
42
(9
), 4272
–4282
(2001
).14.
H.
Goldschmidt
, “Integrability criteria for systems of nonlinear partial differential equations
,” J. Diff. Geom.
1
(3–4
), 269
–307
(1967
).15.
M.
Henneaux
, “Equations of motion, commutation relations and ambiguities in the Lagrangian formalism
,” Ann. Phys.
140
(1
), 45
–64
(1982
).16.
P. E.
Hydon
, “Multisymplectic conservation laws for differential and differential-difference equations
,” Proc. R. Soc. London, Ser. A
461
(2058
), 1627
–1637
(2005
).17.
M.
Juráš
, “The inverse problem of the calculus of variations for sixth- and eighth-order scalar ordinary differential equations
,” Acta Appl. Math.
66
(1
), 25
–39
(2001
).18.
I.
Kolář
, P. W.
Michor
, and J.
Slovák
, Natural Operations in Differential Geometry
(Springer-Verlag
, Berlin
, 1993
).19.
J.
Lee
and R. M.
Wald
, “Local symmetries and constraints
,” J. Math. Phys.
31
(3
), 725
–743
(1990
).20.
P. J.
Olver
, Applications of Lie Groups to Differential Equations
, 2nd ed. Graduate Texts in Mathematics
Vol. 107
(Springer-Verlag
, New York
, 1993
).21.
D. J.
Saunders
, “Thirty years of the inverse problem in the calculus of variations
,” Rep. Math. Phys.
66
(1
), 43
–53
(2010
).22.
B. S. W.
Schröder
, Ordered Sets: An Introduction
(Birkhäuser Boston Inc.
, Boston, MA
, 2003
).23.
Symmetries and Conservation Laws for Differential Equations of Mathematical Physics
, Translations of Mathematical Monographs
Vol. 182
, edited by A. M.
Vinogradov
and I. S.
Krasilshchik
(American Mathematical Society
, Providence, RI
, 1999
).24.
E. P.
Wigner
, “Do the equations of motion determine the quantum mechanical commutation relations?
,” Phys. Rev.
77
, 711
–712
(1950
).25.
G. J.
Zuckerman
, “Action principles and global geometry
,” in Mathematical aspects of string theory
, Advanced Series in Mathematical Physics
, edited by S. T.
Yau
(World Scientific
, Singapore
, 1987
), pp. 259
–284
.26.
Actually,
$\protect \mathrm{d}_{\protect \mathsf {h}}\sigma$
could also be replaced by any merely closed form. However, the cohomology groups H*, v(F) for v > 0 are always trivial (Chap. 5 of Ref. 3), so there is no loss in generality.27.
A preorder P on a set X is a relation such that is reflexive (xPx) and transitive (xPy and yPz implies xPz), but in general neither symmetric nor antisymmetric. All partial orders and equivalence relations are preorders. The maximum symmetric subrelation E (xEy iff xPy and yPx) is an equivalence relation and the quotient X → X/E projects P to a partial order P/E on X/E (which is necessarily antisymmetric).22
© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.
