高阶变分的Hessian态射和Jacobi态射
The Hessian and Jacobi Morphisms for Higher Order Calculus of Variations
摘要：
我们制定高阶变化的拉格朗日在几何框架内的喷流的纤维流形。我们的形式主义适用于拉格朗日依赖于任意数量的独立和从属变量，以及高阶导数。特别是，我们表明，第二个变化是相等的（水平微分）的欧拉-拉格朗日态射的垂直微分原来是自伴沿欧拉-拉格朗日方程的解决方案。这两个对象，分别以不变的方式推广的Hessian态射和Jacobi态射（这是自伴沿临界截面）的情况下，一个给定的拉格朗日高阶拉格朗日。经典拉格朗日函数的一些例子来说明我们的方法。
英文摘要：
We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler--Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.
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