In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.

在本文中，我们提出了在场论中有效的牛顿第二定律的自然概括，即当参数化曲线被更高维的参数化子流形取代时。为此，我们引入了我们称之为测地线k向量场，类似于普通测地线场，它描述了惯性运动（即在没有力的情况下的演化）。根据这个广义牛顿定律，可以通过简单的程序获得相应的哈密顿场论正则方程（Hamilton-De Donder-Weyl 方程）。结果表明，广义牛顿方程的解也成立正则方程。然而，与普通情况不同的是，由不同力确定的牛顿方程可以定义相等的哈密顿方程。