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In 1883, as an early result, Sophus Lie established an explicit necessary andsufficient condition for an analytic second order ordinary differentialequation 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 secondorder analytic ordinary differential equations, was parachieved later in 1896by Arthur Tresse, a French student of S. Lie. In the present paper, followingclosely the original strategy of proof of S. Lie, which we firstly expose andrestitute in length, we generalize this explicit characterization to the caseof several second order ordinary differential equations. Let K=R or C, or moregenerally any field of characteristic zero equipped with a valuation, so thatK-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 manalytic second order ordinary differential equations, in general nonlinear. Weprovide an explicit necessary and sufficient condition in order that thissystem 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 weobtain is of first order in the case m > 1, whereas it is of second order in S.Lie's original case m = 1.Comment: 76 pages, no figur

Characterization of the Newtonian Free Particle System in $m\geqslant 2$ Dependent Variables