Newton's Second Law in Field Theory

Abstract: 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.