Abstract
of H K = 0 be a self-adjoint system. In ?I systems of adjoint differential equations of the second order are discussed. The unique existence of an adjoint system is proved and necessary and sufficient conditions for a given system of the second order to be selfadjoint are established. Characteristic properties of the differential equations of the calculus of variations are derived in ?II. It is here shown that if two equations H = K = 0 are to be the Euler equations for an integral I of the form given above they must have their equations of variation self-adjoint. The sufficiency of this property is established in ?III. Necessary and sufficient conditions for the equations of variation of the system H = K = 0 to be self-adjoint are given. Finally, the most general form of the integrand f for the integral I whose Euler equations are the equations H = K =0 is determined.
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