# nLab de Donder-Weyl formalism

under construction

### Context

#### Variational calculus

variational calculus

# Contents

## Idea

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

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

${\partial }_{\mu }\frac{\delta L}{\delta \left({\partial }_{\mu }{\varphi }^{i}\right)}=\frac{\delta L}{\delta {\varphi }^{i}}$\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?

$\frac{\delta H}{\delta {\pi }_{i}^{\mu }}={\partial }_{\mu }{\varphi }^{i}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\frac{\delta H}{\delta {\varphi }^{i}}=-{\partial }_{\mu }{\pi }_{i}^{\mu }\phantom{\rule{thinmathspace}{0ex}}.$\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

$\Omega :=d{\pi }_{i}^{\mu }\wedge d{\varphi }^{i}\wedge \left({\iota }_{{\partial }_{\mu }}{\mathrm{vol}}_{\Sigma }\right)+dH\wedge {\mathrm{vol}}_{\Sigma }$\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 $\Omega$.

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Reviews include