Explicit differential characterization of the Newtonian free particle system in m > 1 dependent variables

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In 1883, as an early result, Sophus Lie established an explicit necessary and sufficient condition for an analytic second order ordinary differential equation yₓx = F(x,y,yₓ) to be equivalent, through a point transformation (x,y) --> (X(x,y), Y(x,y)), to the Newtonian free particle equation YₓX = 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:= (y¹,..., yᵐ) in Kᵐ and let yₓxʲ = Fʲ(x,y,yₓˡ), 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, y¹,..., yᵐ) --> (X(x,y), Y¹(x,y),..., Yᵐ(x, y)), to the Newtonian free particle system YₓX¹ =... = YₓXᵐ = 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.