We characterize the second variation of an higher order Lagrangian by a Jacobi morphism and by currents strictly related to the geometric structure of the variational problem. We discuss the relation between the Jacobi morphism and the Hessian at an arbitrary order. Furthermore, we prove that a pair of Jacobi fields always generates a (weakly) conserved current. An explicit example is provided for a Yang–Mills theory on a Minkowskian background.

中文翻译：

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高阶场论中的雅可比态射和Hessian；在闵可夫斯基背景下应用杨-米尔斯理论

我们通过雅可比态射和与变分问题的几何结构严格相关的电流来表征高阶拉格朗日的第二变分。我们以任意顺序讨论 Jacobi 态射和 Hessian 之间的关系。此外，我们证明了一对雅可比场总是产生一个（弱）守恒电流。为基于闵可夫斯基背景的杨-米尔斯理论提供了一个明确的例子。