We present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing explicit expressions for the Poisson bracket between two Poisson forms.
REFERENCES
1.
2.
M.
Forger
, C.
Paufler
, and H.
Römer
, Rev. Math. Phys.
15
, 705
(2003
).3.
J.
Kijowski
, Commun. Math. Phys.
30
, 99
(1973
).4.
J.
Kijowski
and W.
Szczyrba
, “Multisymplectic manifolds and the geometrical construction of the Poisson brackets in the classical field theory
,” in Géometrie Symplectique et Physique Mathématique
, edited by J.-M.
Souriau
(CNRS
, Paris, 1975
), pp. 347
–379
.5.
J. F.
Cariñena
, M.
Crampin
, and L. A.
Ibort
, Diff. Geom. Applic.
1
, 345
(1991
).6.
M. J.
Gotay
, “A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism
,” in Mechanics, Analysis and Geometry: after Lagrange, edited by M.
Francaviglia
and D. D.
Holm
, 1991
, pp. 203
–235
.7.
8.
W. M.
Tulczyew
, Bull. Acad. Pol. Sci., Ser. Sci., Math., Astron. Phys.
22
, 937
(1974
).9.
For the sake of brevity, we have left out straightforward but tedious parts of the proofs. The reader may find more detailed guidelines in the arXiv version of this article under math-ph∕0407057v1.
10.
R.
Bryant
(private communication).11.
I. V.
Kanatchikov
, Rep. Math. Phys.
40
, 225
(1997
).12.
C.
Crnković
and E.
Witten
, “Covariant description of canonical formalism in geometrical theories
,” in Three Hundred Years of Gravitation
, edited by W.
Israel
and S.
Hawking
(Cambridge University Press
, Cambridge, 1987
), pp. 676
–684
.13.
C.
Crnković
, Class. Quantum Grav.
5
, 1557
(1988
).14.
G.
Zuckerman
, “Action principles and global geometry
,” in Mathematical Aspects of String Theory
, edited by S.-T.
Yau
(World Scientific
, Singapore, 1987
), pp. 259
–288
.15.
S. V.
Romero
, Ph.D. thesis, IME-USP, 2001
.16.
M.
Forger
and S. V.
Romero
, Commun. Math. Phys.
256
, 375
(2005
).17.
M. O.
Salles
, Ph.D. thesis, IME-USP, 2004
.© 2005 American Institute of Physics.
2005
American Institute of Physics
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