nLab Lepage form

Idea

Lepage forms are certain differential forms on jet bundles. When of maximal horizontal degree then they may be identified with sums $\mathbf{L} + \theta$ of a local Lagrangian $\mathbf{L}$ with its induced presymplectic potential current $\theta$ (as discussed at covariant phase space).

Lepage forms play a central role in the theory of variational sequences.

Definition

Definition

Given a smooth bundle $E \to \Sigma$, then a differential n-form $\rho$ on its jet bundle $Jet(E)$ is called a Lepage form if for every vector field $v$ on the jet bundle, the horizontal part $h(\iota_v \mathbf{d}\rho)$ (with respect to the variational bicomplex) of the contraction of $v$ into the de Rham differential of $\rho$ depends only on the projection of $v$ to a vector field on $E$ itself.

e.g. GMS 09, 2.1.2

In variational calculus

Let $\Sigma$ be of dimension $n$. Then a horizontal $n$-form $\mathbf{L} \in \Omega^{n,0}(Jet(E))$, may be regarded a local Lagrangian. Its de Rham differential has a unique decomposition

$$d \mathbf{L} = \mathbf{E} - d_H \theta$$

where $E$ is a source form, the Euler-Lagrange form of $\mathbf{L}$, and $\theta$, the induced presymplectic potential current, is defined up to addition of a horizontally exact form (e.g. Zuckerman 87, section 2).

Then the combination

$$\rho \coloneqq \mathbf{L} + \theta$$

is a Lepage form, and in fact such a form being Lepage is equivalent to $d \mathbf{L} + d_h \theta$ being a source form.

The differential

$$d \rho = \mathbf{E} + \omega$$

is the sum of the Euler-Lagrange form and the presymplectic current density $\omega = d_V \theta$.

Related concepts

• variational sequence

• multisymplectic geometry

• n-plectic geometry

References

Lepage was a student of Théophile de Donder and did his work on the calculus of variations around the 1930s. Only much later, Krupka was responsible for naming Lepage forms around the 1970s.

See for instance

• Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, Advanced classical field theory, World Scientific, 2009

• G. J. Zuckerman, Action principles and global geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. (pdf)

Exposition of variational calculus in terms of jet bundles and Lepage forms and aimed at examples from physics is in

• Jana Musilová, Stanislav Hronek, The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) (doi.org/10.1515/cm-2016-0012)