An alternate view on symmetries of second-order linearisable ordinary differential equations
The authors introduced in a previous paper [J. Nonlinear Math. Phys. 11, No. 3, 399--421 (2004; Zbl 1075.34033)] the concept of Sundman symmetries for ODEs, in particular for second- and third-order ODEs. Given a third-order ODE \[ x''' = h (x , x' , x'' ; t ), \tag{\(*\)} \] where \(x = x(t)\), one can consider the so-called generalized Sundman transformations \[ X = F (x,t) , \;d T = G(x,t) \, d t, \;G (\partial F / \partial t) \not= 0 \] and it may happen that there exists such a transformation which maps (\(*\)) into \(X''' (T) = 0\). In this case, it was shown by the first author et al. [Acta Appl. Math. 76, No. 1, 89--115 (2003; Zbl 1054.34002)] that there exist another generalized Sundman transformation which also linearizes (\(*\)), given by \[ \widetilde{F} = \frac{1}{F} , \;\widetilde{G} = \frac{G}{F^{3/2}}. \] This implies that any equation (\(*\)) which can be brought to the form \(X''' = 0\) is invariant under the transformation \[ F(\widetilde{x} , \widetilde{t} ) = \frac{1}{F(x,t)} , \;G (\widetilde{x} , \widetilde{t} ) \, d \widetilde{t} = \frac{G(x,t)}{[F(x,t)]^{3/2}} \, d t . \] This is called a Sundman symmetry of (\(*\)). Similar constructions hold for second-order equations (and actually for equations of arbitrary order); a complete classification was given previously for Sundman symmetries of second- and third-order ODEs. In the present work, the authors discuss, and compare with Lie-point ones, symmetries of linearizable second-order ODEs. In particular, they show that, while in this case, the Lie-point symmetry group corresponds to the group of projective transformations in the plane with eight parameters (as shown by M. A. Tresse, a student of Lie's, in 1896), the Sundman symmetries are an infinite set of nonlocal transformations.
scientific article; zbMATH DE number 6145532
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