704
10 Variational Problems of Functionals with Vector, Tensor …
It can be seen from the above examples that using Eqs. (10.6.31) and (10.6.32) we
can easily obtain the Euler equations of the specific functional variation problems
and the corresponding natural boundary conditions.
10.6.4 The Euler Equations of the Functionals
with Parallel-Type Inner Product Tensors
and the Hamiltonian Operator Trains
Some kind of permutation of the gradient, divergence and rotation with Hamiltonian operators is called the Hamiltonian operator train or Hamiltonian operator
string. Such as ∇·∇∇×, ∇∇×∇×∇·, ∇·∇∇×∇×, ·∇∇·∇×∇ and ×∇×∇×∇∇·∇
are all the Hamiltonian operator trains. The Hamiltonian operator train acting on the
left-hand side of a tensor is called the left Hamiltonian operator train, it is written
as ∇ t or ∇ T . Similarly, the Hamiltonian operator train acting on the right-hand side
of a tensor is called the right Hamiltonian operator train, it is written as t ∇ or T ∇.
The left first term of the Hamiltonian operator train is laid on the right first place, The
left second term of the Hamiltonian operator train is laid on the right second place, the
rest can be done in the same manner, until the right-hand side term of the Hamiltonian
operator train is transformed to the most left-hand side. The transformed Hamiltonian
operator train is called the transposition of the Hamiltonian operator train, it is
written as ∇
T
t or ∇
T
T . For example, the transposition of the Hamiltonian operator train
∇∇ × ∇ × ∇· is ∇ · ∇ × ∇ × ∇, the two Hamiltonian operator trains are transposed
to each other; Moreover ∇ × ∇ × ∇ · ∇ is the transformed Hamiltonian operator
train of ∇∇ · ∇ × ∇×, the two Hamiltonian operator trains are also transposed to
each other. The gradient operators and the divergence operators in the Hamiltonian
operator train are transposed, but the rotation operators remain unchanged, such a
transformed Hamiltonian operator train is called the conjugate Hamiltonian operator train or adjoint Hamiltonian operator train, it is written as ∇
c
t or ∇
C
T , ∇
a
t or
∇
A
T etc. For an adjoint Hamiltonian operator train, its sign is such determined: When
the sum of the gradient operator quantities and the divergence operator quantities in
the Hamiltonian operator train is an even integer, the sign is positive, otherwise it is
negative. Such as ∇ · ∇∇× is the adjoint Hamiltonian operator train of ∇ × ∇ · ∇,
they are adjoint to each other; −∇ · ∇∇ × ∇· is the adjoint Hamiltonian operator
train of ∇∇ × ∇ · ∇, they are also adjoint to each other. Every Hamiltonian operator
train has its conjugate Hamiltonian operator train and adjoint Hamiltonian operator
train. If ∇ t = ∇
a
t , then ∇ t is called the self-conjugate Hamiltonian operator train
or self-adjoint Hamiltonian operator train. Obviously, the Laplace operator train
n and rotation operator strain (∇×)
n are both self-conjugate Hamiltonian operator
train or self-adjoint Hamiltonian operator train, of course they are also self-adjoint
operator.
10 Variational Problems of Functionals with Vector, Tensor …
It can be seen from the above examples that using Eqs. (10.6.31) and (10.6.32) we
can easily obtain the Euler equations of the specific functional variation problems
and the corresponding natural boundary conditions.
10.6.4 The Euler Equations of the Functionals
with Parallel-Type Inner Product Tensors
and the Hamiltonian Operator Trains
Some kind of permutation of the gradient, divergence and rotation with Hamiltonian operators is called the Hamiltonian operator train or Hamiltonian operator
string. Such as ∇·∇∇×, ∇∇×∇×∇·, ∇·∇∇×∇×, ·∇∇·∇×∇ and ×∇×∇×∇∇·∇
are all the Hamiltonian operator trains. The Hamiltonian operator train acting on the
left-hand side of a tensor is called the left Hamiltonian operator train, it is written
as ∇ t or ∇ T . Similarly, the Hamiltonian operator train acting on the right-hand side
of a tensor is called the right Hamiltonian operator train, it is written as t ∇ or T ∇.
The left first term of the Hamiltonian operator train is laid on the right first place, The
left second term of the Hamiltonian operator train is laid on the right second place, the
rest can be done in the same manner, until the right-hand side term of the Hamiltonian
operator train is transformed to the most left-hand side. The transformed Hamiltonian
operator train is called the transposition of the Hamiltonian operator train, it is
written as ∇
T
t or ∇
T
T . For example, the transposition of the Hamiltonian operator train
∇∇ × ∇ × ∇· is ∇ · ∇ × ∇ × ∇, the two Hamiltonian operator trains are transposed
to each other; Moreover ∇ × ∇ × ∇ · ∇ is the transformed Hamiltonian operator
train of ∇∇ · ∇ × ∇×, the two Hamiltonian operator trains are also transposed to
each other. The gradient operators and the divergence operators in the Hamiltonian
operator train are transposed, but the rotation operators remain unchanged, such a
transformed Hamiltonian operator train is called the conjugate Hamiltonian operator train or adjoint Hamiltonian operator train, it is written as ∇
c
t or ∇
C
T , ∇
a
t or
∇
A
T etc. For an adjoint Hamiltonian operator train, its sign is such determined: When
the sum of the gradient operator quantities and the divergence operator quantities in
the Hamiltonian operator train is an even integer, the sign is positive, otherwise it is
negative. Such as ∇ · ∇∇× is the adjoint Hamiltonian operator train of ∇ × ∇ · ∇,
they are adjoint to each other; −∇ · ∇∇ × ∇· is the adjoint Hamiltonian operator
train of ∇∇ × ∇ · ∇, they are also adjoint to each other. Every Hamiltonian operator
train has its conjugate Hamiltonian operator train and adjoint Hamiltonian operator
train. If ∇ t = ∇
a
t , then ∇ t is called the self-conjugate Hamiltonian operator train
or self-adjoint Hamiltonian operator train. Obviously, the Laplace operator train
n and rotation operator strain (∇×)
n are both self-conjugate Hamiltonian operator
train or self-adjoint Hamiltonian operator train, of course they are also self-adjoint
operator.