协变经典场论

本条目记法和射流丛条目所引入的一致。并令${\displaystyle {\bar {\Gamma }}(\pi )}$表示有紧支撑的${\displaystyle \pi \,}$的截面。

一个经典场论数学上可以如下表述

• 一个纤维丛 ${\displaystyle ({\mathcal {E}},\pi ,{\mathcal {M}})}$，其中${\displaystyle {\mathcal {M}}}$表示一个${\displaystyle n\,}$维时空。
• 一个拉格朗日量形式 ${\displaystyle \Lambda :J^{1}\pi \rightarrow \Lambda ^{n}M}$

令${\displaystyle \star 1\,}$代表${\displaystyle M\,}$上的体积形式，则${\displaystyle \Lambda =L\star 1\,}$，其中${\displaystyle L:J^{1}\pi \rightarrow \mathbb {R} }$是拉格朗日量函数。 我们在 ${\displaystyle J^{1}\pi \,}$上选择纤维化坐标${\displaystyle \{x^{i},u^{\alpha },u_{i}^{\alpha }\}\,}$，使得

${\displaystyle \star 1=dx^{1}\wedge \ldots \wedge dx^{n}}$

作用量积分定义为

${\displaystyle S(\sigma )=\int _{\sigma ({\mathcal {M}})}(j^{1}\sigma )^{*}\Lambda \,}$

其中${\displaystyle \sigma \in {\bar {\Gamma }}(\pi )}$，并定义于开集${\displaystyle \sigma ({\mathcal {M}})\,}$，而${\displaystyle j^{1}\sigma \,}$代表其第一射流延长(jet prolongation)。

截面${\displaystyle \sigma \in {\bar {\Gamma }}(\pi )\,}$的变分由曲线${\displaystyle \sigma _{t}=\eta _{t}\circ \sigma \,}$给出，其中${\displaystyle \eta _{t}\,}$是一个${\displaystyle {\mathcal {E}}\,}$上的${\displaystyle \pi \,}$-竖直向量场${\displaystyle V\,}$的流，它在${\displaystyle {\mathcal {M}}\,}$上有紧支撑。 截面${\displaystyle \sigma \in {\bar {\Gamma }}(\pi )\,}$称为变分的驻点，如果

${\displaystyle \left.{\frac {d}{dt}}\right|_{t=0}\int _{\sigma ({\mathcal {M}})}(j^{1}\sigma _{t})^{*}\Lambda =0\,}$

这等价于

${\displaystyle \int _{\mathcal {M}}(j^{1}\sigma )^{*}{\mathcal {L}}_{V^{1}}\Lambda =0\,}$

其中${\displaystyle V^{1}\,}$代表${\displaystyle V\,}$的第一延长，按李导数的定义。 使用嘉当公式，${\displaystyle {\mathcal {L}}_{X}=i_{X}d+di_{X}\,}$， 斯托克斯定理以及${\displaystyle \sigma \,}$的紧支撑，可以证明这等价于

${\displaystyle \int _{\mathcal {M}}(j^{1}\sigma )^{*}i_{V^{1}}d\Lambda =0\,}$

考虑一个${\displaystyle {\mathcal {E}}}$的${\displaystyle \pi \,}$-竖直向量场

${\displaystyle V=\beta ^{\alpha }{\frac {\partial }{\partial u^{\alpha }}}\,}$

其中${\displaystyle \beta ^{\alpha }=\beta ^{\alpha }(x,u)\,}$。采用切触形式 ${\displaystyle \theta ^{j}=du^{j}-u_{i}^{j}dx^{i}\,}$ on ${\displaystyle J^{1}\pi \,}$，我们可以计算${\displaystyle V\,}$的第一延长。然后得到

${\displaystyle V^{1}=\beta ^{\alpha }{\frac {\partial }{\partial u^{\alpha }}}+\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}u_{i}^{j}\right){\frac {\partial }{\partial u_{i}^{\alpha }}}\,}$

其中${\displaystyle \gamma _{i}^{\alpha }=\gamma _{i}^{\alpha }(x,u^{\alpha },u_{i}^{\alpha })\,}$。 据此，可以证明

${\displaystyle i_{V^{1}}d\Lambda =\left[\beta ^{\alpha }{\frac {\partial L}{\partial u^{\alpha }}}+\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}u_{i}^{j}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\right]\star 1\,}$

因而

${\displaystyle (j^{1}\sigma )^{*}i_{V^{1}}d\Lambda =\left[(\beta ^{\alpha }\circ \sigma ){\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}\circ \sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}\circ \sigma \right){\frac {\partial \sigma ^{j}}{\partial x^{i}}}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right]\star 1\,}$

分部积分并考虑${\displaystyle \sigma \,}$的紧支撑，临界条件变为

${\displaystyle \int _{\mathcal {M}}(j^{1}\sigma )^{*}i_{V^{1}}d\Lambda \,}$	${\displaystyle =\int _{\mathcal {M}}\left[{\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma -{\frac {\partial }{\partial x^{i}}}\left({\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right)\right](\beta ^{\alpha }\circ \sigma )\star 1\,}$
	${\displaystyle =0\,}$

因为${\displaystyle \beta ^{\alpha }\,}$为任意函数，我们得到

${\displaystyle {\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma -{\frac {\partial }{\partial x^{i}}}\left({\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right)=0\,}$

这些就是欧拉－拉格朗日方程组。

• 经典场论
• 外代数
• 纤维丛
• 射流
• 量子场论

• 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