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EXPLICIT DIFFERENTIAL CHARACTERIZATION OF THE NEWTONIAN FREE PARTICLE SYSTEM IN m ≥ 2 DEPENDENT VARIABLES
"... ABSTRACT. 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 eq ..."
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ABSTRACT. 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 grouptheoretic 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 Kanalytic functions make sense. Let x ∈ K, let m ≥ 2, let y: = (y 1,..., y m) ∈ K m and let y 1 xx = F 1 (x, y, yx) ,......, y m xx = F m (x, y, yx), 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, y 1,..., y m) ↦ → ( X(x, y), Y 1 (x, y),..., Y m (x, y) ), to the Newtonian free particle system Y 1 XX = · · · = Y m XX = 0. Strikingly, the (complicated) differential system that we obtain is of first order in the case m ≥ 2, whereas it is of second order in S. Lie’s original case m = 1. Table of contents
Explicit differential characterization of PDE systems pointwise equivalent to Y X j 1
 X j 2 = 0, 1 ≤ j1, j2 ≤ n ≥ 2, in preparation. JOËL MERKER
"... 1 ≤ j1, j2 ≤ n ≥ 2. ..."
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FOUR EXPLICIT FORMULAS FOR THE PROLONGATIONS OF AN INFINITESIMAL LIE SYMMETRY AND MULTIVARIATE FAÀ DI BRUNO FORMULAS
, 2004
"... ABSTRACT. In 1979, building on S. Lie’s theory of symmetries of (partial) differential equations, P.J. Olver formulated inductive formulas which are appropriate for the computation of the prolongations of an infinitesimal Lie symmetry to jet spaces, for an arbitrary number n ≥ 1 of independent varia ..."
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ABSTRACT. In 1979, building on S. Lie’s theory of symmetries of (partial) differential equations, P.J. Olver formulated inductive formulas which are appropriate for the computation of the prolongations of an infinitesimal Lie symmetry to jet spaces, for an arbitrary number n ≥ 1 of independent variables (x 1,..., x n) and for an arbitrary number m ≥ 1 of dependent variables (y 1,..., y m). This paper is devoted to elaborate a formalism based on multiple Kronecker symbols which enables one to handle these “unmanageable ” prolongations and to discover the underlying complicated combinatorics. Proceeding progressively, we write down closed explicit
On the analyticity . . .
, 2014
"... In any positive CRdimension and CRcodimension we provide a construction of realanalytic holomorphically nondegenerate CRsubmanifolds, which are C∞ CRequivalent, but are inequivalent holomorphically. As a corollary, we provide the negative answer to the conjecture of Ebenfelt and Huang [20] on t ..."
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In any positive CRdimension and CRcodimension we provide a construction of realanalytic holomorphically nondegenerate CRsubmanifolds, which are C∞ CRequivalent, but are inequivalent holomorphically. As a corollary, we provide the negative answer to the conjecture of Ebenfelt and Huang [20] on the analyticity of CRequivalences between realanalytic Levi nonflat hypersurfaces in dimension 2.