nLab
Lepage form

Contents

Contents

Idea

Lepage forms are certain differential forms on jet bundles. When of maximal horizontal degree then they may be identified with sums L+θ\mathbf{L} + \theta of a local Lagrangian L\mathbf{L} with its induced presymplectic potential current θ\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 EΣE \to \Sigma, then a differential n-form ρ\rho on its jet bundle Jet(E)Jet(E) is called a Lepage form if for every vector field vv on the jet bundle, the horizontal part h(ι vdρ)h(\iota_v \mathbf{d}\rho) (with respect to the variational bicomplex) of the contraction of vv into the de Rham differential of ρ\rho depends only on the projection of vv to a vector field on EE itself.

e.g. GMS 09, 2.1.2

In variational calculus

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

dL=Ed Hθ d \mathbf{L} = \mathbf{E} - d_H \theta

where EE is a source form, the Euler-Lagrange form of L\mathbf{L}, and θ\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

ρL+θ \rho \coloneqq \mathbf{L} + \theta

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

The differential

dρ=E+ω d \rho = \mathbf{E} + \omega

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

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

Last revised on February 16, 2017 at 07:52:39. See the history of this page for a list of all contributions to it.