An alternative approach for the analysis and the numerical
approximation of ODEs, using a variational framework, is
presented. It is based on the natural and elementary idea of minimizing
the residual of the differential equation measured in a usual Lp
Typical existence results for Cauchy problems can thus be
recovered, and finer sets of assumptions for existence are made explicit. We treat, in particular, the cases of an explicit ODE and a differential inclusion. This approach also allows for a whole strategy to approximate numerically the solution. It is briefly indicated here as it will be pursued systematically and in a much more broad fashion in a subsequent paper.