含哈密顿算子的并联式内积张量泛函变分问题

Variational Problems of the Functional of Parallel-Type Inner Product Tensor with Hamiltonian Operators

讨论变分法中含哈密顿算子即梯度、散度和旋度的并联式张量的泛函变分问题.根据n阶张量并联式内积和串联式内积运算规则,给出张量泛函变分问题的基本引理.提出并证明含哈密顿算子的张量泛函变分问题的定理;通过直接对张量的梯度、散度和旋度进行变分,得到欧拉方程和相应的自然边界条件.通过若干算例验证了欧拉方程的正确性.扩展伴随算子的内涵,提出右伴随算子的概念,讨论伴随算子或自伴算子与梯度、散度和旋度算子的关系,指出所讨论的泛函变分问题实质上是符合伴随算子或自伴算子定义的运算.

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.