Variational problems with lags

In this paper, we study some questions concerning the minima of the functional $$J\left( y \right) = \int_{x_1 }^{x_2 } {f\left( {x,y\left( x \right),y\left( {x - r} \right),\dot y\left( x \right),\dot y\left( {x - r} \right)} \right)dx} $$ In Section 2, we obtain an analogue to the Jacobi condition to add to the list of previously obtained necessary conditions. A transversality condition is developed in Section 3. In Section 4, we obtain an existence theorem. The techniques used are modifications of those used in the classical problems. In Section 5, we show that the theory of fields for the classical problem fails to work for our problem.