Covariant classical field theory

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In recent years, there has been renewed interest in covariant classical field theory. Here, dynamics are phrased in the context of a finite-dimensional space of fields at a given event in spacetime. Nowadays, it is well known that jet bundles are the correct domain for such a description. This article provides some of the geometric structure to the covariant formalism of first-order classical field theories.

Contents

1 Notation
2 The Action Integral
3 Variation of the Action Integral
4 The Euler-Lagrange Equations
5 See also
6 References

Notation

The notation follows that of introduced in the article on jet bundles. Also, let [\bar(\pi)] denote the set of sections of [\pi\,] with compact support.

The Action Integral

A classical field theory is mathematically described by

A fibre bundle [(\mathcal,\pi, \mathcal)], where [\mathcal] denotes an [n\,]-dimensional spacetime.
A Lagrangian form [\Lambda:J^\pi \rightarrow \Lambda^M]

Let [\star 1\,] denote the volume form on [M\,], then [\Lambda = L\star 1\,] where [L:J^\pi \rightarrow \mathbb] is the Lagrangian function. We choose fibred co-ordinates [\,u^,u^_\}\,] on [J^\pi\,], such that

[\star 1 = dx^ \wedge \ldots \wedge dx^]

The action integral is defined by

[S(\sigma) = \int_)} (j^\sigma)^\Lambda \,]

where [\sigma \in \bar(\pi)] and is defined on an open set [\sigma(\mathcal)\,], and [j^\sigma\,] denotes its first jet prolongation.

Variation of the Action Integral

The variation of a section [\sigma \in \bar(\pi)\,] is provided by a curve [\sigma_ = \eta_ \circ \sigma\,], where [\eta_\,] is the flow of a [\pi\,]-vertical vector field [V\,] on [\mathcal\,], which is compactly supported in [\mathcal\,]. A section [\sigma \in \bar(\pi)\,] is then stationary with respect to the variations if

[\left.\frac\right|_\int_)}(j^\sigma_)^\Lambda = 0\,]

This is equivalent to

[\int_} (j^\sigma)^\mathcal_}\Lambda = 0\,]

where [V^\,] denotes the first prolongation of [V\,], by definition of the Lie derivative. Using Cartan's formula, [\mathcal_=i_d + di_\,], Stokes' Theorem and the compact support of [\sigma\,], we may show that this is equivalent to

[\int_} (j^\sigma)^i_}d\Lambda = 0 \,]

The Euler-Lagrange Equations

Considering a [\pi\,]-vertical vector field on [\mathcal]

[V = \beta^\frac}\,]

where [\beta^ = \beta^(x,u)\,]. Using the contact forms [\theta^ = du^ - u^_dx^\,] on [J^\pi\,], we may calculate the first prolongation of [V\,]. We find that

[V^ = \beta^\frac} + \left(\frac}} + \frac}}u^_\right)\frac_}\,]

where [\gamma^_ = \gamma^_(x,u^,u^_)\,]. From this, we can show that

[i_}d\Lambda = \left[beta^frac} + left(frac}} + frac}}u^_right)frac_}right]\star 1 \,]

and hence

[(j^\sigma)^i_}d\Lambda = \left[(beta^ circ sigma)frac} circ j^sigma + left(frac}} circ sigma + left(frac}} circ sigma right)frac}} right)frac_} circ j^sigma right]\star 1 \,]

Integrating by parts and taking into account the compact support of [\sigma\,], the criticality condition becomes

[\int_} (j^\sigma)^i_}d\Lambda \,] [= \int_} \left[frac} circ j^sigma - frac} left(frac_} circ j^sigma right)right]( \beta^\circ \sigma )\star 1 \,]
[= 0 \,]

and since the [\beta^\,] are arbitrary functions, we obtain

[\frac} \circ j^\sigma - \frac} \left(\frac_} \circ j^\sigma \right) = 0\,]

These are the Euler-Lagrange Equations.

References

Saunders, D.J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
Bocharov, A.V. [et al.] "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958
De Leon, M., Rodrigues, P.R., "Generalized Classical Mechanics and Field Theory", Elsevier Science Publishing, 1985, ISBN 0-444-87753-3
Griffiths, P.A., "Exterior Differential Systems and the Calculus of Variations", Boston: Birkhauser, 1983, ISBN 3-764-33103-8
Gotay, M.J., Isenberg, J., Marsden, J.E., Montgomery R., [Momentum Maps and Classical Fields Part I: Covariant Field Theory], November 2003
Echeverria-Enriquez, A., Munoz-Lecanda, M.C., Roman-Roy,M., [Geometry of Lagrangian First-order Classical Field Theories], May 1995

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[\int_} (j^\sigma)^i_}d\Lambda \,]	[= \int_} \left[frac} circ j^sigma - frac} left(frac_} circ j^sigma right)right]( \beta^\circ \sigma )\star 1 \,]
	[= 0 \,]