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 equivalents associated with a local Lagrangian in particular allow to get the Euler-Lagrange form (which vanishes along the pullback of any section that satisfies the Euler-Lagrange equation) from its de Rham differential, thus avoiding the need to use the Euler operator?.

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 $\omega = d_V \theta$.

To obtain the Euler-Lagrange form one needs to apply an operator, called the Euler operator? to $\mathbf{L}$. In contrast, the same Euler-Lagrange form may be obtained from $\mathbf{L}+\theta$ by applying the usual de Rham operator. This serves as a motivation to replace the Lagrangian $\mathbf{L}$ with the form $\mathbf{L}+\theta$. Consistent replacements are known as Lepage equivalents.

Definition

Let $E \to \Sigma$ be a smooth bundle with $\Sigma$ of dimension $n$. Given a horizontal $n$-form $\mathbf{L} \in \Omega^{n,0}(Jet(E))$, regarded as a local Lagrangian , a Lepage equivalent $\rho_L$ is a $n$-form $\rho_L\in\Omega^n(Jet(E))$ (not necessarily horizontal) such that a. The horizontal component of $\rho_L$ is $mathbf{L}$, and b. The (n,1)-component of $d \rho_L$ is the Euler-Lagrange form $E$ of $\mathbf{L}$.

Lepage equivalents are rarely unique, and many times fail to be globally-defined even if the Lagrangian itself is globally defined. A notable exception is classical mechanics with $F$ a finite-dimensional smooth manifold and order-1 Lagrangian (meaning it depends on at most first-order derivatives fields), for which there is a unique Lepage equivalent, called the Poincaré-Cartan form.

Note that the definition of Lepage equivalent only involves restrictions on the horizontal and 1-vertical components. This means that given a Lepage equivalent $\rho_L$, one can shift by forms of bidegree $(n-k,k)$ for $k\geq 2$ and still obtain a Lepage equivalent even though this affects the resulting pre-$n$-plectic form. One way towards addressing this problem is to demand a further condition called closure property , which removes the contributions from trivial Lagrangians.

Definition

A Lepage equivalent $\rho_L$ is said to satisfy the closure property if whenever the Euler-Lagrange form $E$ of the Lagrangian $\mathbf{L}$ is identically zero, then $d\rho_L=0$ vanishes identically too.

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, Demeter 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)

• Demeter Krupka. Introduction to global variational geometry. Vol. 1. Amsterdam: Atlantis Press, (2015). (doi:10.2991/978-94-6239-073-7)