  In 1883, as an early result, Sophus Lie established an explicit necessary and
sufficient condition for an analytic second order ordinary differential
equation y_xx = F(x,y,y_x) to be equivalent, through a point transformation
(x,y) --> (X(x,y), Y(x,y)), to the Newtonian free particle equation Y_XX = 0.
This result, preliminary to the deep group-theoretic classification of second
order analytic ordinary differential equations, was parachieved later in 1896
by Arthur Tresse, a French student of S. Lie. In the present paper, following
closely the original strategy of proof of S. Lie, which we firstly expose and
restitute in length, we generalize this explicit characterization to the case
of several second order ordinary differential equations. Let K=R or C, or more
generally any field of characteristic zero equipped with a valuation, so that
K-analytic functions make sense. Let x in K, let m > 1, let y := (y^1, ...,
y^m) in K^m and let y_xx^j = F^j(x,y,y_x^l), j = 1,...,m be a collection of m
analytic second order ordinary differential equations, in general nonlinear. We
provide an explicit necessary and sufficient condition in order that this
system is equivalent, under a point transformation (x, y^1, ..., y^m) -->
(X(x,y), Y^1(x,y),..., Y^m(x, y)), to the Newtonian free particle system Y_XX^1
= ... = Y_XX^m = 0. Strikingly, the (complicated) differential system that we
obtain is of first order in the case m > 1, whereas it is of second order in S.
Lie's original case m = 1.
