nLab
de Donder-Weyl formalism

under construction

Idea

The formalism of de Donder-Weyl is a formulation of variational calculus related to multisymplectic geometry.

For $L$ a local Lagrangian of fields $\{{\varphi }^i\}$ on a space $\Sigma$, and their first derivatives ${\partial }_{\mu }{\varphi }^i$, the Euler-Lagrange equations

$${\partial }_{\mu }\frac{\delta L}{\delta ({\partial }_{\mu }{\varphi }^i)}=\frac{\delta L}{\delta {\varphi }^i}$$\partial_\mu \frac{\delta L }{\delta (\partial_\mu \phi^i)} = \frac{\delta L}{ \delta \phi^i}

are equivalently rewritten, under the generalized Legendre transform (if it exists)

$$H:={\pi }_i^{\mu }{\partial }_{\mu }{\varphi }^i-L$$H := \pi^\mu_i \partial_\mu \phi^i - L

(with ${\pi }_i^{\mu }$ the ”$\mu$-th momentum of the $i$-th field” ) as the first-order system of “covariant” Hamilton equations?

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

This $H$ is also called the de Donder-Weyl Hamiltonian.

In terms of the multisymplectic form

$$\Omega :=d{\pi }_i^{\mu }\wedge d{\varphi }^i\wedge ({\iota }_{{\partial }_{\mu }}{\mathrm{vol}}_{\Sigma })+dH\wedge {\mathrm{vol}}_{\Sigma }$$\Omega := d \pi^\mu_i \wedge d \phi^i \wedge (\iota_{\partial_\mu} vol_\Sigma) + d H \wedge vol_\Sigma

the critical field configurations are precisely those whose top-degree multi-tangent spaces annihilate $\Omega$.

(…)

References

Reviews include

• 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

• 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.