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The de Donder-Weyl-Hamilton equation is a refinement of Hamilton's equation from a single variational direction to many.
Where Hamilton's equation appears as a natural condition in symplectic geometry, so the de Donder-Weyl-Hamilton equation appears as the analogous condition in multisymplectic geometry/n-plectic geometry.
Where Hamilton's equation appears as the equations of motion of a mechanical system whose dynamics is described by evolution along a single parameter (time), so the de Donder-Weyl Hamilton equation appears as the equations of motion of a higher dimensional field theory given by a local Lagrangian where “evolution” is along more parameters (spacetime).
Let $\Sigma$ be a smooth manifold to be regarded as worldvolume/spacetime and let $F \to \Sigma$ be a smooth bundle to be thought of as the field bundle.
For simplicity of exposition we first consider local patches and take without essential restriction $\Sigma$ to be a Cartesian space and the field bundle to be a trivial vector bundle $X \times \Sigma$. In all of the following the summation convention? for summation over repeated induces is understood.
We then denote the canonical coordinates on $\Sigma$ by $\{\sigma^\mu\}$ and those on $X$ by $\{\phi^i\}$.
Write $(J^1 E^)^\ast$ for the dualized first jet bundle of $E$. Under the above assumptions this has canonical local coordinates $\{\sigma^\mu, \phi^i, \pi_i^\mu\}$. Here we call $\pi^\mu_i$ the “$\mu$-th momentum of the $i$-th field”.
Consider a local Lagrangian $L$ which is of first order, hence which is a function on the first jet bundle $J^1 E$ of the field bundle.
Its Euler-Lagrange equations are
The de Donder-Weyl-Hamiltonian of $L$ is its generalized Legendre transform (if it exists), the function
on the dual jet bundle.
In terms of the de Donder-Weyl Hamiltonian $H$, def. 2, the Euler-Lagrange equations of motion, def. 1, are equivalent to the differential equations
These are called the de Donder-Weyl-Hamilton equations.
For $(X,\omega)$ a symplectic manifold and $H \in C^\infty(X)$ a smooth function regarded as a Hamiltonian, then Hamilton's equations are equivalent to
for $v$ a tangent vector to a trajectory in phase space. This may be referred to as the “non-relativistic” form of the symplectic version of Hamilton’s equations, as the “time”-parameter is not part of the phase space.
By passing from plain phase space to the corresponding dual jet bundle, hence by adjoining a worldline coordinate $t$, this is equivalent to
where now
This may accordingly be thought of as the relativistic version of Hamilton’s equations.
We now discuss the analog of both the “non-relativistic” and of this “relativistic version” of Hamilton's equation for the de Donder-Weyl-Hamilton equation.
In both cases, the $n$-tuple of tangent vectors to a section which satisfies the equations of motion is characterized as a Hamiltonian n-vector field. See there for more discussion.
Given a de Donder-Weyl-Hamiltonian $H$, def. 2, define a differential form
on the dual jet bundle (the multisymplectic form or pre-(n+1)-plectic form) by
A section $(\phi^i, \pi^\mu_i) \colon \Sigma \to J^1 E^\ast$ of the dualized first jet bundle satisfies the equations of motion, prop. 1, precisely if its tangent vectors
jointly annihilate the pre-$(n+1)$-plectic form of def. 3 in that the equation:
holds.
First, the component of this equation which does not contain any $\mathbf{d}\sigma^\mu$ is
This is already equivalent to the DWH equation, prop. 1.
Second, this already implies that the components of the equation that are proportional to $\mathbf{d}\sigma^\mu$ are automatically satisfied; because these components are
and inserting the DWH equations on the left makes the left side identically equal to the right hand side.
Proposition 2 is indeed true in the general case where $H$ may be spacetime dependent (depend nontrivially on the $\sigma^\mu$).
To obtain the “non-relativistic” form of the $(n+1)$-plectic form of the DWH equation, consider the affine dual first jet bundle with canonical coordinates $\{\sigma^a , \phi^i, \pi^a_i, e \}$.
The canonical pre-(n+1)-plectic form on the affine dual first jet bundle is
Vector fields tangent to a section of the affine dual first jet bundle are of the form
On the affine dual first jet bundle the de Donder-Weyl-Hamilton equation characterizes those sections whose tangent vectors as above satisfy
This has been pointed out in (Hélein 02, around equation (4)).
This is a slight variant of the proof of prop. 2. First, the component of the equation independend of $\mathbf{d}\sigma^\nu$ is
Here the term $\mathbf{d}e$ cancels on both sides and leaves the equation equivalent to the DWH equation as in the first step of the proof of prop. 2.
Second, the component of the claimed equation proportional to $\mathbf{d}\sigma^\mu$ is now
Inserting into this the DWH equation makes it equivalent to
This has a unique solution, up to a global constant, given by
By the proof of prop. 3 we have on shell that $H + e = 0$ and that $\omega = \Omega$.
Maybe the first example of what is now called De Donder-Weyl theory appeared in
Then Weyl and de Donder independently published
Hermann Weyl, Geodesic fields in the calculus of variations, Ann. Math. (2) 36 (1935) 607-629.
Théophile de Donder, Théorie invariante du calcul des variations, Nuov. éd, Gauthiers–Villars, Paris 1935
Reviews include
Wikipedia, De Donder-Weyl theory
Narciso Román-Roy, Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories, SIGMA 5 (2009), 100 (journal, arXiv:math-ph/0506022)
Frédéric Hélein, Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory, (arXiv:math-ph/0212036)
See also
For more see the references at multisymplectic geometry, at n-plectic geometry and at Hamiltonian n-vector field.