The main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values. We review the concept of Type-I/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi's solution to the Hamilton–Jacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the so-called multisymplectic form formula. As an example of the latter, we show that Lorentz's reciprocity principle in electromagnetism is a particular instance of the multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges.
REFERENCES
1.
R.
Abraham
and J. E.
Marsden
, Foundations of Mechanics
, 2nd ed. (revised and enlarged) (Benjamin/Cummings Publishing Co. Inc., Advanced Book Program
, Reading, MA
, 1978
) (with the assistance of T. Raţiu and R. Cushman).2.
D. N.
Arnold
, R. S.
Falk
, and R.
Winther
, “Finite element exterior calculus: From Hodge theory to numerical stability
,” Bull. Am. Math. Soc.
47
(2
), 281
–354
(2010
).3.
David E.
Betounes
, “Extension of the classical Cartan form
,” Phys. Rev. D
29
(4
), 599
–606
(1984
).4.
E.
Binz
, J.
Śniatycki
, and H.
Fischer
, Geometry of Classical Fields
, North-Holland Mathematics Studies, Notas de Matemática [Mathematical Notes]
Vol. 154
(North-Holland Publishing Co.
, Amsterdam
, 1988
), p. 123
.5.
A.
Bossavit
, Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements
(Academic Press Inc.
, San Diego, CA
, 1998
).6.
T. J.
Bridges
, “Multi-symplectic structures and wave propagation
,” Math. Proc. Cambridge Philos. Soc.
121
(1
), 147
–190
(1997
).7.
T. J.
Bridges
and S.
Reich
, “Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity
,” Phys. Lett. A
284
(4–5
), 184
–193
(2001
).8.
T. J.
Bridges
and S.
Reich
, “Numerical methods for Hamiltonian PDEs
,” J. Phys. A
39
(19
), 5287
–5320
(2006
).9.
F.
Cantrijn
, A.
Ibort
, and M.
de León
, “On the geometry of multisymplectic manifolds
,” J. Aust. Math. Soc. Ser. A
66
(3
), 303
–330
(1999
).10.
J. F.
Cariñena
, M.
Crampin
, and L. A.
Ibort
, “On the multisymplectic formalism for first order field theories
,” Dif. Geom. Applic.
1
(4
), 345
–374
(1991
).11.
P. R.
Chernoff
and J. E.
Marsden
, Properties of Infinite Dimensional Hamiltonian Systems
, Lecture Notes in Mathematics
Vol. 425
(Springer-Verlag
, Berlin
, 1974
).12.
Č.
Crnković
and E.
Witten
, “Covariant description of canonical formalism in geometrical theories
,” Three Hundred Years of Gravitation
(Cambridge University Press
, Cambridge
, 1987
), pp. 676
–684
.13.
P. L.
García
, “The Poincaré-Cartan invariant in the calculus of variations
,” Simposia Mathematica (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973)
(Academic Press
, London
, 1974
), Vol. XIV
, pp. 219
–246
.14.
M.
Gotay
, J.
Isenberg
, and J. E.
Marsden
, “Momentum maps and classical relativistic fields. Part I: Covariant field theory
,” preprint arXiv:physics/9801019 (1997
).15.
M.
Gotay
, J.
Isenberg
, and J. E.
Marsden
, “Momentum maps and classical relativistic fields. Part II: Canonical analysis of field theories
,” preprint arXiv:math-ph/0411032 (1999
).16.
A. N.
Hirani
, “Discrete exterior calculus
,” Ph.D. thesis (California Institute of Technology
, 2003
).17.
D. D.
Holm
, R. I.
Ivanov
, and J. R.
Percival
, “G. Strands
,” J. Nonlinear Sci.
22
(4
), 517
–551
(2012
).18.
J.-M.
Jin
, Theory and Computation of Electromagnetic Fields
(Wiley
, 2010
).19.
J.
Kijowski
and W. M.
Tulczyjew
, A Symplectic Framework for Field Theories
, Lecture Notes in Physics
Vol. 107
(Springer-Verlag
, Berlin
, 1979
).20.
S.
Kouranbaeva
and S.
Shkoller
, “A variational approach to second-order multisymplectic field theory
,” J. Geom. Phys.
35
(4
), 333
–366
(2000
).21.
B.
Lawruk
, J.
Śniatycki
, and W. M.
Tulczyjew
, “Special symplectic spaces
,” J. Differ. Equations
17
, 477
–497
(1975
).22.
M.
Leok
, “Foundations of computational geometric mechanics
,” Ph.D. thesis (California Institute of Technology
, 2004
).23.
M.
Leok
and T.
Shingel
, “General techniques for constructing variational integrators
,” Front. Math. China
7
, 273
–303
(2012
).24.
M.
Leok
and J.
Zhang
, “Discrete Hamiltonian variational integrators
,” IMA J. Numer. Anal.
31
(4
), 1497
–1532
(2011
).25.
J. E.
Marsden
and M.
West
, “Discrete mechanics and variational integrators
,” Acta Numer.
10
, 357
–514
(2001
).26.
J. E.
Marsden
, G. W.
Patrick
, and S.
Shkoller
, “Multisymplectic geometry, variational integrators, and nonlinear PDEs
,” Commun. Math. Phys.
199
(2
), 351
–395
(1998
).27.
C. W.
Misner
, K. S.
Thorne
, and J. A.
Wheeler
, Gravitation
(W. H. Freeman and Co.
, San Francisco, CA
, 1973
).28.
T.
Ohsawa
, A. M.
Bloch
, and M.
Leok
, “Discrete Hamilton–Jacobi theory
,” SIAM J. Control Optim.
49
(4
), 1829
–1856
(2011
).29.
C.
Rovelli
, Quantum Gravity
, Cambridge Monographs on Mathematical Physics (Cambridge University Press
, Cambridge
, 2004
) (with a foreword by J. Bjorken).30.
J.
Śniatycki
and W.
Tulczyjew
, “Generating forms of lagrangian submanifolds
,” Indiana Univ. Math. J.
22
, 267
–275
(1973
).31.
A.
Stern
, Y.
Tong
, M.
Desbrun
, and J. E.
Marsden
, “Variational integrators for Maxwell's equations with sources
,” PIERS Online
4
(7
), 711
–715
(2008
) (previously appeared in PIERS Proceedings, Cambridge, USA, July 2–6, 2008, pp. 443–447).32.
J.
Vankerschaver
and F.
Cantrijn
, “Discrete Lagrangian field theories on Lie groupoids
,” J. Geom. Phys.
57
(2
), 665
–689
(2007
).33.
R. M.
Wald
, General Relativity
(University of Chicago Press
, Chicago, IL
, 1984
).34.
G. J.
Zuckerman
, “Action principles and global geometry
,” Mathematical Aspects of String Theory
, Advanced Series in Mathematical Physics
Vol. 1
(World Scientific Publishing
, Singapore
, 1987
), pp. 259
–284
.35.
It is interesting to note from a historical point of view that a precursor of the multisymplectic form formula already appears in the seminal work of
R.
Courant
, K.
Friedrichs
, and H.
Lewy
, “Über die partiellen Differenzengleichungen der mathematischen Physik
,” Math. Ann.
100
(1
), 32
–74
(1928
).© 2013 AIP Publishing LLC.
2013
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