nLab
source form

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Definition

Given a smooth bundle EΣE \to \Sigma, then a differential form on its jet bundle J Σ (E)J^\infty_\Sigma(E) is called a source form if it is of vertical degree 1 (with respect to the variational bicomplex) and its evaluation on a vector field depends only on the projection of that vector field to a vector field on EE itself (e.g. Zuckerman 87, p. 6):

Ω Σ,source k,1(E)Ω Σ k,0(E)δC (E). \Omega^{k,1}_{\Sigma,source}(E) \;\coloneqq\; \Omega^{k,0}_\Sigma(E) \wedge \delta C^\infty(E) \,.

In variational calculus

Given a local Lagrangian L\mathbf{L}, i.e. a horizontal form on Jet(E)Jet(E) of maximal horizontal degree, then there is a unique source form E\mathbf{E} such that

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

for some form θ\theta. This E\mathbf{E} is the Euler-Lagrange form of L\mathbf{L}.

The sum

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

is the corresponding Lepage form.

References

  • 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)

Last revised on November 30, 2017 at 07:34:07. See the history of this page for a list of all contributions to it.