Some nonlinear planar sprays (Q2711854)

A (general) spray on a manifold \(M\) is defined as a projectable section \(\xi\) of the second order tangent bundle \(TTM\) of \(M\). \textit{W. Ambrose, R. S. Palais} and \textit{I. M. Singer} [An. Acad. Brasil. Ci. 32, 163-178 (1960; Zbl 0097.37904)] introduced sprays (for the geometric description of second-order differential equations on manifolds) as a curve in \(M\) is an integral of \(\xi\) if and only if it is a solution of the second order equation: \({d^2x^i}/{dt^2}=\psi^i(x,{dx}/{dt})\). The authors give a summary of previous results [Geom. Dedicata 55, 211-220 (1995; Zbl 0834.53032)] and numerous examples for second order differential equations on the plane. They give a lot of figures to illustrate the examples.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00012].