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10 Variational Problems of Functionals with Vector, Tensor …
10.7.3 The Euler Equations of the Functionals
with Series-Type Inner Product Tensors
and the Hamiltonian Operator Trains
Taking the tensor of the Hamiltonian operator train acting on as the axis of symmetry,
the Hamiltonian operator train obtained after taking the mirror image of the Hamiltonian operator train is called the mirror image Hamiltonian operator train, it is
denoted by ∇
m
t or ∇
M
T etc. First taking the mirror image to the Hamiltonian operator
train, and then taking its transposition, or first taking transposition to the Hamiltonian
operator train, and then taking its mirror image, such the obtained Hamiltonian operator train is called the mirror image transposition Hamiltonian operator train or
the transposition mirror image Hamiltonian operator train. The gradient operators and divergence operators in the mirror image transposition Hamiltonian operator
train are transposed, but the rotation operators remain unchanged; In other words, for
a Hamiltonian operator train, transpose ∇ and × in the rotation operators, transpose
∇ and • in the divergence operators, change the gradient operators into the divergence operators, and change the divergence operators into the gradient operators;
Such the obtained Hamiltonian operator train is called the mirror image conjugate
Hamiltonian operator train or the conjugate mirror image Hamiltonian operator train, or the mirror image adjoint Hamiltonian operator train, or the adjoint
mirror image Hamiltonian operator train, it is called the conjugate Hamiltonian
operator train or adjoint Hamiltonian operator train for short, they are denoted
by ∇
c
t or ∇
C
T etc. For an adjoint Hamiltonian operator train, its sign is determined
like this: When the sum of the Hamiltonian operator quantities in the Hamiltonian
operator train is an even integer, the sign is positive, otherwise it is negative. Such as,
×∇ × ∇∇ · ∇ is the mirror image Hamiltonian operator train of ∇ · ∇∇ × ∇×, and
∇ · ∇ × ∇ × ∇ is the adjoint Hamiltonian operator train of ∇ · ∇∇ × ∇×; ×∇∇ · ∇
is the mirror image Hamiltonian operator train of ∇ · ∇∇×, and −∇ · ∇ × ∇ is the
adjoint Hamiltonian operator train of ∇ · ∇∇×. Every Hamiltonian operator train
has its mirror transposition Hamiltonian operator train and mirror image conjugate
Hamiltonian operator train.
Similarly, using the concepts of the above mentioned Hamiltonian operator strains,
the more complicated variational problem can also be discussed.
Theorem 10.7.2 Let A be a tensor of order n, ∇ t is the left Hamiltonian operator
train,
c
t ∇ is the conjugate Hamiltonian operator train of ∇ t , t ∇ is the right Hamiltonian
operator train, ∇
c
t is the conjugate Hamiltonian operator train of t ∇, the functional
with the tensor and the Hamiltonian operator trains can be expressed as
J =
V
F( A, A
T
, tr A, ∇ t A t ∇)dV
(10.7.30)
then, the Euler equation of the functional is
10 Variational Problems of Functionals with Vector, Tensor …
10.7.3 The Euler Equations of the Functionals
with Series-Type Inner Product Tensors
and the Hamiltonian Operator Trains
Taking the tensor of the Hamiltonian operator train acting on as the axis of symmetry,
the Hamiltonian operator train obtained after taking the mirror image of the Hamiltonian operator train is called the mirror image Hamiltonian operator train, it is
denoted by ∇
m
t or ∇
M
T etc. First taking the mirror image to the Hamiltonian operator
train, and then taking its transposition, or first taking transposition to the Hamiltonian
operator train, and then taking its mirror image, such the obtained Hamiltonian operator train is called the mirror image transposition Hamiltonian operator train or
the transposition mirror image Hamiltonian operator train. The gradient operators and divergence operators in the mirror image transposition Hamiltonian operator
train are transposed, but the rotation operators remain unchanged; In other words, for
a Hamiltonian operator train, transpose ∇ and × in the rotation operators, transpose
∇ and • in the divergence operators, change the gradient operators into the divergence operators, and change the divergence operators into the gradient operators;
Such the obtained Hamiltonian operator train is called the mirror image conjugate
Hamiltonian operator train or the conjugate mirror image Hamiltonian operator train, or the mirror image adjoint Hamiltonian operator train, or the adjoint
mirror image Hamiltonian operator train, it is called the conjugate Hamiltonian
operator train or adjoint Hamiltonian operator train for short, they are denoted
by ∇
c
t or ∇
C
T etc. For an adjoint Hamiltonian operator train, its sign is determined
like this: When the sum of the Hamiltonian operator quantities in the Hamiltonian
operator train is an even integer, the sign is positive, otherwise it is negative. Such as,
×∇ × ∇∇ · ∇ is the mirror image Hamiltonian operator train of ∇ · ∇∇ × ∇×, and
∇ · ∇ × ∇ × ∇ is the adjoint Hamiltonian operator train of ∇ · ∇∇ × ∇×; ×∇∇ · ∇
is the mirror image Hamiltonian operator train of ∇ · ∇∇×, and −∇ · ∇ × ∇ is the
adjoint Hamiltonian operator train of ∇ · ∇∇×. Every Hamiltonian operator train
has its mirror transposition Hamiltonian operator train and mirror image conjugate
Hamiltonian operator train.
Similarly, using the concepts of the above mentioned Hamiltonian operator strains,
the more complicated variational problem can also be discussed.
Theorem 10.7.2 Let A be a tensor of order n, ∇ t is the left Hamiltonian operator
train,
c
t ∇ is the conjugate Hamiltonian operator train of ∇ t , t ∇ is the right Hamiltonian
operator train, ∇
c
t is the conjugate Hamiltonian operator train of t ∇, the functional
with the tensor and the Hamiltonian operator trains can be expressed as
J =
V
F( A, A
T
, tr A, ∇ t A t ∇)dV
(10.7.30)
then, the Euler equation of the functional is