Mechanical Systems with Random Perturbations

Abstract

It is a well-known fact that a second order differential equation $\ddot{x}(t)=\bar{\alpha}(t,x(t),\dot{x}(t))$ expressing Newton’s law in ℝⁿ may be represented as a first order system on the space of dimension 2n:

$$ \left\{\begin{array}{rcl}\dot{x}(t) & = & v(t) \\\dot{v}(t) & = & \bar{\alpha}(t,x(t),v(t)).\end{array}\right.$$

(1)

We call the first equation of the above system horizontal and the second one vertical. This is consistent with the terminology in the general case of a mechanical system on a non-linear configuration space M, where Newton’s law is presented by means of covariant derivatives in the form (11.2), equivalent to equation (11.3) with a special vector field (second order differential equation) on the phase space TM. Recall that the field (11.3) is the sum of the Levi-Civitá geodesic spray $\mathcal{Z}$ (which is horizontal, i.e. belongs to the connection) and the vertical lift of the vector force field (which belongs to the vertical subspace).