Abstract
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
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
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 \((m=1)\).
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Merker, J. Characterization of the Newtonian Free Particle System in \(m\geqslant 2\) Dependent Variables. Acta Appl Math 92, 125–207 (2006). https://doi.org/10.1007/s10440-006-9064-z
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DOI: https://doi.org/10.1007/s10440-006-9064-z