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with a given system of °o 4 curves defined by the differential equations .. y" = 4>(x,y,z,y',z')f z" = yp(x,y,z,y',z'), we must find F so as to satisfy simultaneously Fy Fyx y'Fyy z'Fy. tpFyy pFy,. = 0, (o) F. F,.x y'F,,y z'F,>, <pFyt. pp.,,, = 0. This can not be done, in general. Thus a system of extremals of a problem of the form (4) must have properties which distinguish it in the totality of curve families definable by equations of the form (5). To express these

Extremals and transversality of the general calculus of variations problem of the first order in space