Since Fermat also, we know that the points at which ~ achieves its extremums are critical points of 9. Thus, any way which succeeds in proving, directly, that ~ has a maximum or a minimum provides a way of proving the existence of a solution of (i). This is the so-called direct method of the calculus of variations which goes back to Gauss, Kelvin, Dirichlet, Hilbert, Tonelli and others. More recent work deals with Droving the existence of critical points at which ~ does not achieve an extremum (saddle points). This paper surveys some of the recent work in this direction. A systematic exposition of many aspects of the variational approach to boundary-value problems for ordinary differential equations will be given in [11].