Infinite-dimensional second order ordinary differential equations via T2MT2M

The vector bundle structure obtained on the second order (acceleration) tangent bundle T2MT2M of a smooth manifold MM by means of a linear connection on the base provides an alternative way for the study of second order ordinary differential equations on manifolds of finite and infinite dimension. Second order vector fields and their integral curves could provide a new way of representing and solving a wide class of evolutionary equations for states on Fréchet manifolds of sections that arise naturally as inequivalent configurations of a physical field. The technique is illustrated by examples in the framework of Banach and Fréchet spaces, and on Lie groups, in particular discussing the case of autoparallel curves, which include geodesics if the connection is induced by a Riemannian structure.