nLab De Donder-Weyl-Hamilton equation

Idea

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).

Definition

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.

Definition

Its Euler-Lagrange equations are

$$\partial_\mu \frac{\delta L }{\delta (\partial_\mu \phi^i)} = \frac{\delta L}{ \delta \phi^i} \,.$$

Definition

The De Donder-Weyl-Hamiltonian of $L$ is its generalized Legendre transform (if it exists), the function

$$H \coloneqq \pi^\mu_i \partial_\mu \phi^i - L$$

on the dual jet bundle.

Proposition/Definition

In terms of the De Donder-Weyl Hamiltonian $H$, def. , the Euler-Lagrange equations of motion, def. , are equivalent to the differential equations

$$\partial_\mu \phi^i = \frac{\partial H}{\partial \pi_i^\mu} \;\;\,, \;\;\;\; \partial_\mu \pi^\mu_i = - \frac{\partial H}{ \partial \phi^i} \,.$$

These are called the De Donder-Weyl-Hamilton equations.

Properties

As a multisymplectic/$n$-plectic Hamiltonian equation

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

$$\iota_v \omega = \mathbf{d}H$$

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

$$\iota_v \Omega = 0$$

where now

$$\Omega = \omega + \mathbf{d}H \wedge \mathbf{d}t \,.$$

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.

Relativistic form

Definition

Given a De Donder-Weyl-Hamiltonian $H$, def. , define a differential form

$$\Omega \in \Omega^{n+1}_{cl}((J^1 E)^\ast)$$

on the dual jet bundle (the multisymplectic form or pre-(n+1)-plectic form) by

$$\Omega \;\coloneqq \; \mathbf{d} \phi^i \wedge \mathbf{d} \pi^\mu_i \wedge (\iota_{\partial_\mu} vol_\Sigma) + \mathbf{d} H \wedge vol_\Sigma \,.$$

Proposition

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. , precisely if its tangent vectors

$$v_\mu \coloneqq \partial_{\mu} + \frac{\partial \phi^i}{\partial \sigma^\mu} \partial_{\phi^i} + \frac{\partial \pi_i^\nu}{\partial \sigma^\mu} \partial_{p^\nu_i}$$

jointly annihilate the pre-$(n+1)$-plectic form of def. in that the equation:

$$(\iota_{v_1} \cdots \iota_{v_n}) \Omega = 0$$

holds.

Proof

First, the component of this equation which does not contain any $\mathbf{d}\sigma^\mu$ is

$$\iota_{v_i} \; \mathbf{d}\phi^i \wedge \mathbf{d}\pi^i_\mu = \mathbf{d}H \,.$$

This is already equivalent to the DWH equation, prop. .

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

$$\frac{\partial \phi^i}{\partial \sigma^\mu} \frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu - \frac{\partial \phi^i}{\partial \sigma^\nu} \frac{\partial \pi_i^\mu}{\partial \sigma^\mu} \mathbf{d}\sigma^\nu = \frac{\partial H}{\partial \phi^i}\frac{\partial \phi^i}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu + \frac{\partial H}{\partial \pi_i^\mu}\frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu$$

and inserting the DWH equations on the left makes the left side identically equal to the right hand side.

Remark

Proposition is indeed true in the general case where $H$ may be spacetime dependent (depend nontrivially on the $\sigma^\mu$).

Non-relativistic form

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 $\{\sigma^a , \phi^i, \pi^a_i, e \}$.

Remark

The canonical pre-(n+1)-plectic form on the affine dual first jet bundle is

$$\omega \coloneqq \mathbf{d}\phi^i \wedge \mathbf{d}\phi^\mu_a \iota_{\partial_{\sigma^\mu}} vol_\Sigma + \mathbf{d}e \wedge vol_\Sigma \,.$$

Remark

Vector fields tangent to a section of the affine dual first jet bundle are of the form

$$v_\mu \coloneqq \partial_{\mu} + \frac{\partial \phi^i}{\partial \sigma^\mu} \partial_{\phi^i} + \frac{\partial \pi_i^\nu}{\partial \sigma^\mu} \partial_{p^\nu_i} + \frac{\partial e}{\partial \sigma^\mu} \partial_e \,.$$

Proposition

On the affine dual first jet bundle the de Donder-Weyl-Hamilton equation characterizes those sections whose tangent vectors as above satisfy

$$(\iota_{v_1} \cdots \iota_{v_n}) \omega = \mathbf{d}(H + e)$$

This has been pointed out in (Hélein 02, around equation (4)).

Proof

This is a slight variant of the proof of prop. . First, the component of the equation independend of $\mathbf{d}\sigma^\nu$ is

$$\iota_{v_i} \; \mathbf{d}\pi^i_\mu \wedge \mathbf{d}\phi^i + \mathbf{d}e = \mathbf{d}H + \mathbf{d}e \,.$$

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. .

Second, the component of the claimed equation proportional to $\mathbf{d}\sigma^\mu$ is now

$$\frac{\partial \phi^i}{\partial \sigma^\mu} \frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu - \frac{\partial \phi^i}{\partial \sigma^\nu} \frac{\partial \pi_i^\mu}{\partial \sigma^\mu} \mathbf{d}\sigma^\nu + \frac{\partial e}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu = 0 \,.$$

Inserting into this the DWH equation makes it equivalent to

$$\frac{\partial H}{\partial \phi^i}\frac{\partial \phi^i}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu + \frac{\partial H}{\partial \pi_i^\mu}\frac{\partial \pi_i^\mu}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu + \frac{\partial e}{\partial \sigma^\nu} \mathbf{d}\sigma^\nu = 0$$

This has a unique solution, up to a global constant, given by

$$e(\{ \sigma^\nu \}) = - H( \{ \phi^i(\{ \sigma^\nu \}), \pi^\mu_i (\{ \sigma^\nu \}) \} ) \,.$$

Remark

By the proof of prop. we have on shell that $H + e = 0$ and that $\omega = \Omega$.

Related concepts

• Hamiltonian n-vector field

References

Maybe the first example of what is now called De Donder-Weyl theory appeared in

• Constantin Carathéodory, Über die Extremalen und geodätischen Felder in der Variationsrechnung der mehrfachen Integrale, Acta Sci. Math. (Szeged) 4 (1929) 193-216

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:

• 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

• Igor V. KanatchikovGeometric (pre)quantization in the polysymplectic approach to field theory, in Differential Geometry and Its Applications, Proc. Conf., Opava (August 2001) [arXiv:hep-th/0112263, pdf]

• Igor V. Kanatchikov, De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory, Rept. Math. Phys. 43 (1999) 157-170 [arXiv:hep-th/9810165, doi:10.1016/S0034-4877(99)80024-X]

For teleparallel gravity:

• Igor V. Kanatchikov. The De Donder-Weyl Hamiltonian formulation of TEGR and its quantization (2023) [arXiv:2308.10052]

For more see the references at multisymplectic geometry, at n-plectic geometry and at Hamiltonian n-vector field.