Acta Applicandae Mathematica

, Volume 92, Issue 2, pp 125–207

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


DOI: 10.1007/s10440-006-9064-z

Cite this article as:
Merker, J. Acta Appl Math (2006) 92: 125. doi:10.1007/s10440-006-9064-z


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 \((m=1)\).

Key words

general relativity Hamiltonian dynamics Lie Newtonian mechanics. 

Mathematics Subject Classifications (2000)

Primary 58F36 Secondary 34A05, 58A15, 58A20, 58F36, 34C14, 32V40. 

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.CNRS, Université de Provence, LATPMarseille Cedex 13France

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