10.6 Variational Problems of Functionals with Parallel-Type Inner …
685
10.6 Variational Problems of Functionals
with Parallel-Type Inner Product Tensors
and Hamiltonian Operators
In mathematics, physics, theory of elasticity, thermodynamics, fluid mechanics,
material science and other some subject areas, the functionals with the tensors and
Hamiltonian operators would be often met, the solutions of the variational problems
of this kind of the functional are the main points researched in the natural science,
engineering and technology, therein an important step is to determine the Euler equation of the functional. But how to determine the Euler equation of the functional is a
quite difficult problem, it is usually for some concrete problem that start from definition of the functional variation the corresponding Euler equation is derived, such
derivation process is tedious detail, it is very difficult to do the work. The variational
problems of the functionals with the tensors of order n and the Hamiltonian operators
namely with the gradient, divergence and rotation are discussed in the present section,
through directly applying the variation to the gradients, divergences and rotations of
the tensor, the corresponding Euler equations and the natural boundary conditions
are obtained, some famous examples are given by means of the obtained Euler equations, therefore the correctness of the Euler equations are verified. The concepts of
Hamiltonian operator trains are presented, based on these concepts the more complicated variational problems are discussed, the corresponding Euler equations and the
natural boundary conditions are given.
10.6.1 Variational Formula Derivations of Gradients,
Divergences and Rotations of Parallel-Type Inner
Product Tensors
Let A be a tensor of order n, A = A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n , D is a tensor of order n − 1,
D = D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1 , take ∂ i =
∂
∂ x i
, ∇ = ∂ i e i =
∂
∂ x i
e i , do n−1 time inner
product of A and D, then take the divergence, note that A n − 1
· D is a vector, and for a
vector a, there is ∇ · a = a ·∇, according to the comma convention, F ,i =
∂ F
∂ x i
= ∂ i F,
there is
∇ · ( A n − 1
· D) = ( A n − 1
· D) · ∇
= ∂ i e i · (A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n
n − 1
· D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1 )
= (A j 1 j 2 ··· j n ,i D k 1 k 2 ···k n−1
+ A j 1 j 2 ··· j n D k 1 k 2 ···k n−1 ,i )(e i · e j 1 )(e j 2 · e k 1 ) · · · (e j n · e k n−1 )
= A j 1 j 2 ··· j n ,i D k 1 k 2 ···k n−1 (e i · e j 1 )(e j 2 · e k 1 ) · · · (e j n · e k n−1 )
685
10.6 Variational Problems of Functionals
with Parallel-Type Inner Product Tensors
and Hamiltonian Operators
In mathematics, physics, theory of elasticity, thermodynamics, fluid mechanics,
material science and other some subject areas, the functionals with the tensors and
Hamiltonian operators would be often met, the solutions of the variational problems
of this kind of the functional are the main points researched in the natural science,
engineering and technology, therein an important step is to determine the Euler equation of the functional. But how to determine the Euler equation of the functional is a
quite difficult problem, it is usually for some concrete problem that start from definition of the functional variation the corresponding Euler equation is derived, such
derivation process is tedious detail, it is very difficult to do the work. The variational
problems of the functionals with the tensors of order n and the Hamiltonian operators
namely with the gradient, divergence and rotation are discussed in the present section,
through directly applying the variation to the gradients, divergences and rotations of
the tensor, the corresponding Euler equations and the natural boundary conditions
are obtained, some famous examples are given by means of the obtained Euler equations, therefore the correctness of the Euler equations are verified. The concepts of
Hamiltonian operator trains are presented, based on these concepts the more complicated variational problems are discussed, the corresponding Euler equations and the
natural boundary conditions are given.
10.6.1 Variational Formula Derivations of Gradients,
Divergences and Rotations of Parallel-Type Inner
Product Tensors
Let A be a tensor of order n, A = A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n , D is a tensor of order n − 1,
D = D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1 , take ∂ i =
∂
∂ x i
, ∇ = ∂ i e i =
∂
∂ x i
e i , do n−1 time inner
product of A and D, then take the divergence, note that A n − 1
· D is a vector, and for a
vector a, there is ∇ · a = a ·∇, according to the comma convention, F ,i =
∂ F
∂ x i
= ∂ i F,
there is
∇ · ( A n − 1
· D) = ( A n − 1
· D) · ∇
= ∂ i e i · (A j 1 j 2 ··· j n e j 1 e j 2 · · · e j n
n − 1
· D k 1 k 2 ···k n−1 e k 1 e k 2 · · · e k n−1 )
= (A j 1 j 2 ··· j n ,i D k 1 k 2 ···k n−1
+ A j 1 j 2 ··· j n D k 1 k 2 ···k n−1 ,i )(e i · e j 1 )(e j 2 · e k 1 ) · · · (e j n · e k n−1 )
= A j 1 j 2 ··· j n ,i D k 1 k 2 ···k n−1 (e i · e j 1 )(e j 2 · e k 1 ) · · · (e j n · e k n−1 )