de Donder-Weyl-Hamilton equation


Variational calculus


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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 Σ\Sigma be a smooth manifold to be regarded as worldvolume/spacetime and let FΣF \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 Σ\Sigma to be a Cartesian space and the field bundle to be a trivial vector bundle X×ΣX \times \Sigma. In all of the following the summation convention? for summation over repeated induces is understood.

We then denote the canonical coordinates on Σ\Sigma by {σ μ}\{\sigma^\mu\} and those on XX by {ϕ i}\{\phi^i\}.

Write (J 1E )*(J^1 E^)^\ast for the dualized first jet bundle of EE. Under the above assumptions this has canonical local coordinates {σ μ,ϕ i,π i μ}\{\sigma^\mu, \phi^i, \pi_i^\mu\}. Here we call π i μ\pi^\mu_i the “μ\mu-th momentum of the ii-th field”.

Consider a local Lagrangian LL which is of first order, hence which is a function on the first jet bundle J 1EJ^1 E of the field bundle.


Its Euler-Lagrange equations are

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

The de Donder-Weyl-Hamiltonian of LL is its generalized Legendre transform (if it exists), the function

Hπ i μ μϕ iL H \coloneqq \pi^\mu_i \partial_\mu \phi^i - L

on the dual jet bundle.


In terms of the de Donder-Weyl Hamiltonian HH, def. 2, the Euler-Lagrange equations of motion, def. 1, are equivalent to the differential equations

μϕ i=Hπ i μ, μπ i μ=Hϕ i. \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.


As a multisymplectic/nn-plectic Hamiltonian equation

For (X,ω)(X,\omega) a symplectic manifold and HC (X)H \in C^\infty(X) a smooth function regarded as a Hamiltonian, then Hamilton's equations are equivalent to

ι vω=dH \iota_v \omega = \mathbf{d}H

for vv 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 tt, this is equivalent to

ι vΩ=0 \iota_v \Omega = 0

where now

Ω=ω+dHdt. \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 nn-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


Given a de Donder-Weyl-Hamiltonian HH, def. 2, define a differential form

ΩΩ cl n+1((J 1E) *) \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

Ωdϕ idπ i μ(ι μvol Σ)+dHvol Σ. \Omega \;\coloneqq \; \mathbf{d} \phi^i \wedge \mathbf{d} \pi^\mu_i \wedge (\iota_{\partial_\mu} vol_\Sigma) + \mathbf{d} H \wedge vol_\Sigma \,.

A section (ϕ i,π i μ):ΣJ 1E *(\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

v μ μ+ϕ iσ μ ϕ i+π i νσ μ p i ν 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)(n+1)-plectic form of def. 3 in that the equation:

(ι v 1ι v n)Ω=0 (\iota_{v_1} \cdots \iota_{v_n}) \Omega = 0



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

ι v idϕ idπ μ i=dH. \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. 1.

Second, this already implies that the components of the equation that are proportional to dσ μ\mathbf{d}\sigma^\mu are automatically satisfied; because these components are

ϕ iσ μπ i μσ νdσ νϕ iσ νπ i μσ μdσ ν=Hϕ iϕ iσ νdσ ν+Hπ i μπ i μσ νdσ ν \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.


Proposition 2 is indeed true in the general case where HH may be spacetime dependent (depend nontrivially on the σ μ\sigma^\mu).

Non-relativistic form

To obtain the “non-relativistic” form of the (n+1)(n+1)-plectic form of the DWH equation, consider the affine dual first jet bundle with canonical coordinates {σ a,ϕ i,π i a,e}\{\sigma^a , \phi^i, \pi^a_i, e \}.


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

ωdϕ idϕ a μι σ μvol Σ+devol Σ. \omega \coloneqq \mathbf{d}\phi^i \wedge \mathbf{d}\phi^\mu_a \iota_{\partial_{\sigma^\mu}} vol_\Sigma + \mathbf{d}e \wedge vol_\Sigma \,.

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

v μ μ+ϕ iσ μ ϕ i+π i νσ μ p i ν+eσ μ e. 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 \,.

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

(ι v 1ι v n)ω=d(H+e) (\iota_{v_1} \cdots \iota_{v_n}) \omega = \mathbf{d}(H + e)

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 dσ ν\mathbf{d}\sigma^\nu is

ι v idπ μ idϕ i+de=dH+de. \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 de\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 dσ μ\mathbf{d}\sigma^\mu is now

ϕ iσ μπ i μσ νdσ νϕ iσ νπ i μσ μdσ ν+eσ νdσ ν=0. \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

Hϕ iϕ iσ νdσ ν+Hπ i μπ i μσ νdσ ν+eσ νdσ ν=0 \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({σ ν})=H({ϕ i({σ ν}),π i μ({σ ν})}). e(\{ \sigma^\nu \}) = - H( \{ \phi^i(\{ \sigma^\nu \}), \pi^\mu_i (\{ \sigma^\nu \}) \} ) \,.

By the proof of prop. 3 we have on shell that H+e=0H + e = 0 and that ω=Ω\omega = \Omega.


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 include

See also

  • I.V. Kanatchikov, DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory , (arXiv:hep-th/9810165)

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

Revised on February 23, 2015 21:04:50 by Urs Schreiber (