含哈密顿算子的并联式内积张量泛函变分问题

The variational problems of the functional depending on parallel-type inner product tensor in the calculus of variations were discussed, the tensor in the functional can include Hamiltonian operator namely the gradient, divergence and curl operator. According to the parallel-type and serial-type inner product operation rules of the n-th order tensor, the fundamental lemma of the variational operations of the functional depending on the tensor was given. The theorem of the variational problem of the functional depending on the tensor with Hamiltonian operators was proposed and proved; Through making directly variation to the tensors with Hamiltonian operators, the Euler equation and the corresponding natural boundary conditions were obtained. The correctness of the Euler equation was verified with some functional examples. The connotation of the dajoint operator was extended, the conception of the right adjoint operator was proposed, the relationships between the adjoint operator or self-adjoint operator and the gradient, divergence and curl operator were discussed, it is pointed out that the variational problem of the functional discussed is essentially operation conforming to the definition of adjoint operator operation or self-adjoint operator.