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arXiv math-ph/0411026math-phmath.DGmath.MP
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规范自然不变拉格朗日量的二阶变分导数与守恒律

Second variational derivative of gauge-natural invariant Lagrangians and conservation laws

M. Francaviglia, M. Palese, E. Winterroth

备注：
17 pages; some misprints corrected, few changes, reference list updated, v3 to appear in Proc. IX Int. Conf. on Diff. Geom. and its Appl. (Prague 30/08-03/09/2004)
发表信息：
Proc.IX Int. Conf. Diff. Geom. Appl.; J.Bures et al. eds.; Charles University, Prague (Czech Republic), 2005, 591--604
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摘要：

我们考虑一个给定的规范自然不变拉格朗日量的第二变分导数的规范自然升降机的无穷小的主自同构的垂直部分（简化）。通过要求这样一个二阶变分导数为零，{ \em via}第二Noether定理，我们发现一个协变的强守恒流与变形的拉格朗日量正则相关，该变形的拉格朗日量是通过将原始拉格朗日量的Euler-Lagrange方程与位于广义规范-自然Jacobi态射核中的无穷小主自同构的规范-自然提升的垂直部分收缩而得到的。

英文摘要：

We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find that a covariant strongly conserved current is canonically associated with the deformed Lagrangian obtained by contracting Euler--Lagrange equations of the original Lagrangian with (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms lying in the kernel of the generalized gauge-natural Jacobi morphism.

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