Characterization of the Newtonian free particle system in \(m\geqslant 2\) dependent variables. (English) Zbl 1330.34016
Summary: 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
\[
y_{xx}^1 = F^1(x,y,y_x), \ldots, y_{xx}^m = F^m(x,y,y_x),
\]
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
\[
(x,y^1,\ldots,y^m) \mapsto \left(X(x,y),Y^1(x,y),\ldots,Y^m(x,y)\right),
\]
to the Newtonian free particle system \(Y^{1}_{XX} = \cdots = Y^{m}_{XX} = 0\).
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 (\(m=1\)).
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 (\(m=1\)).
MSC:
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 34A26 | Geometric methods in ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 37C80 | Symmetries, equivariant dynamical systems (MSC2010) |
| 70H99 | Hamiltonian and Lagrangian mechanics |
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