610
10 Variational Problems of Functionals with Vector, Tensor …
of theorem. Giving some examples of practical problems to verify the correctness
of the Euler equations. Then discuss the more complex variational problems of the
functionals with the gradient, divergence and rotation.
For the variational problem of the functional with parallel type n order tensor,
nth order transposed tensor and Hamiltonian operators, first deduce the variational
formulas of the gradient, divergence and rotation for the nth order tensor respectively, then through the way of merging give the theorem of variational problem of
such functional, obtain the Euler equation and the corresponding natural boundary
conditions, through some actual functionals illustrate the usage of the Euler equation,
consequently to verify the correctness of the Euler equation. Put forward the concept
of Hamiltonian operator string, on the basis discuss the more complex variational
problem of the functional, including the variational problem the trace of tensor, give
the corresponding Euler equation, give the establishing method of natural boundary
conditions. Have a discussion to the other variational problem of the functional with
the parallel type inner product tensor and Hamiltonian operators, present the corresponding Euler equations and natural boundary conditions. In a similar way to discuss
the variational problem of the functional with the series type nth order tensor and
Hamiltonian operators.
Through the variational operation of the functional with the parallel type nth
order tensor, series type nth order tensor and the Hamiltonian operators, reveal the
relationships between the variational operation of the functional with the Hamiltonian
operator and the adjoint operator, illustrate that the four kinds of adjoint operators
defined in Chap. 7 exist and have the practical significance. Pointing out the internal
relations between the four kinds of adjoint operators, in some cases, the four kinds
of adjoint operators are the very self-adjoint operators, and they are equivalent. This
shows that the discussed variational problems of the functional are essentially the
operations conforming to the definitions of the four kinds of adjoint operators or
self-adjoint operators in Hilbert space.
In the process of formulas derivation, this chapter assumes that all operations are
meaningful, the case of operation failure is not considered.
10.1 Basic Properties of the Tensor Inner Product
Operations and Fundamental Lemma of the Variation
of Functional with Tensors
Let A and B be both the tensors of order n presented with the dyads, A = a 1 a 2 · · · a n ,
B = b 1 b 2 · · · b n , then the parallel type n time inner product of A and B has following
operation properties
A n
· B = a 1 a 2 · · · a n−1 a n n
· b 1 b 2 · · · b n−1 b n
= (a 1 · b 1 )(a 2 · b 2 ) · · · (a n−1 · b n−1 )(a n · b n )
10 Variational Problems of Functionals with Vector, Tensor …
of theorem. Giving some examples of practical problems to verify the correctness
of the Euler equations. Then discuss the more complex variational problems of the
functionals with the gradient, divergence and rotation.
For the variational problem of the functional with parallel type n order tensor,
nth order transposed tensor and Hamiltonian operators, first deduce the variational
formulas of the gradient, divergence and rotation for the nth order tensor respectively, then through the way of merging give the theorem of variational problem of
such functional, obtain the Euler equation and the corresponding natural boundary
conditions, through some actual functionals illustrate the usage of the Euler equation,
consequently to verify the correctness of the Euler equation. Put forward the concept
of Hamiltonian operator string, on the basis discuss the more complex variational
problem of the functional, including the variational problem the trace of tensor, give
the corresponding Euler equation, give the establishing method of natural boundary
conditions. Have a discussion to the other variational problem of the functional with
the parallel type inner product tensor and Hamiltonian operators, present the corresponding Euler equations and natural boundary conditions. In a similar way to discuss
the variational problem of the functional with the series type nth order tensor and
Hamiltonian operators.
Through the variational operation of the functional with the parallel type nth
order tensor, series type nth order tensor and the Hamiltonian operators, reveal the
relationships between the variational operation of the functional with the Hamiltonian
operator and the adjoint operator, illustrate that the four kinds of adjoint operators
defined in Chap. 7 exist and have the practical significance. Pointing out the internal
relations between the four kinds of adjoint operators, in some cases, the four kinds
of adjoint operators are the very self-adjoint operators, and they are equivalent. This
shows that the discussed variational problems of the functional are essentially the
operations conforming to the definitions of the four kinds of adjoint operators or
self-adjoint operators in Hilbert space.
In the process of formulas derivation, this chapter assumes that all operations are
meaningful, the case of operation failure is not considered.
10.1 Basic Properties of the Tensor Inner Product
Operations and Fundamental Lemma of the Variation
of Functional with Tensors
Let A and B be both the tensors of order n presented with the dyads, A = a 1 a 2 · · · a n ,
B = b 1 b 2 · · · b n , then the parallel type n time inner product of A and B has following
operation properties
A n
· B = a 1 a 2 · · · a n−1 a n n
· b 1 b 2 · · · b n−1 b n
= (a 1 · b 1 )(a 2 · b 2 ) · · · (a n−1 · b n−1 )(a n · b n )