, Volume 92, Issue 2, pp 125-207
Date: 08 Sep 2006

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

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We treat the problem of linearizability of a system of second order ordinary differential equations. The criterion we provide has applications to nonlinear Newtonian mechanics, especially in three-dimensional space. Let \({\mathbb K}={\mathbb R}\) or \({\mathbb C}\) , let \(x \in {\mathbb K}\) , let \(m\geqslant 2\) , let \(y:=(y^1,\ldots,y^m)\in {\mathbb K}^m\) and let $$y_{xx}^1=F^1\left(x, y, y_x\right),\ldots\dots,y_{xx}^m=F^m\left( x,y,y_x \right),$$ be a collection of m analytic second order ordinary differential equations, in general nonlinear. We obtain a new and applicable necessary and sufficient condition in order that this system is equivalent, under a point transformation $$(x, y^1,\dots, y^m) \mapsto \left( X(x,y), Y^1(x,y),\dots, Y^m(x, y)\right),$$ to the Newtonian free particle system \(Y^{1}_{{XX}} = \dots = Y^{m}_{{XX}} = 0\) .

Strikingly, the explicit differential system that we obtain is of first order in the case \(m\geqslant 2\) , whereas according to a classical result due to Lie, it is of second order the case of a single equation \((m=1)\) .