Complete lift transformations for second order differential equations (Q1910076)

The paper deals with a system modelled by a smooth vector field \(F\) on the state space and an output function \(h\), i.e. \[ \begin{cases} v' & = F(v)\\ w & = h(v)\end{cases} \] where the pair \((F,h)\) is called an observed dynamics. A special case of the observed dynamics, when \(w''= f(w,w')\), is considered in detail. For such a case, several canonical forms are obtained under action of the group of complete lift transformations. Discussions of the results obtained and various illustrative examples are also presented.