  A general variational principle of classical fields with a Lagrangian
containing the field quantity and its derivatives of up to the N-th order is
presented. Noether's theorem is derived. The generalized Hamilton-Jacobi's
equation for the Hamilton's principal functional is obtained. These results are
surprisingly in great harmony with each other. They will be applied to the
general relativity in the subsequent articles, especially the generalized
Noether's theorem will be applied to the problem of conservation and
non-conservation in curved spacetime..
