Chapter 10
Variational Problems of Functionals
with Vector, Tensor and Hamiltonian
Operators
The variational problems of functionals with the scalar, vector, modulus of vector,
tensor, trace of tensor, transposed tensor, Hamiltonian operator and Hamiltonian
operator string are discussed in this chapter, which include the functionals of any
complex form of vectors, nth order tensor and its gradient, divergence and rotation,
for this purpose, a series of new concepts, new theories and new ways of thinking are
advanced, in order to build the science building of the variational operation of the
above mentioned functionals, derive the corresponding Euler equations and natural
boundary conditions. The general method of variational problem of the functional
with Hamiltonian operators has been found, this is taking the divergence operation
of the vector as the starting point. With the methods advanced in this chapter it
can establish an infinite number of Euler equations of the functionals containing the
scalar, vector, tensor and Hamiltonian operators and obtain the corresponding natural
boundary conditions.
According to the need of various variational operations of the functionals with
the tensor, first to introduce the basic properties of the parallel type and series type
two kinds of inner product operations of the two nth order tensors, by the way to give
the variational formula of the nth order transposed tensor. On this basis, advance the
fundamental lemma of the functional variation, which lays a theoretical foundation
for the operation of variational problem of the functional with the vector, tensor and
Hamiltonian operators.
With the development of science and technology and the progress of the society,
more and more research fields involve the variational problems of the functional with
the scalar, vector, modulus of vector and Hamiltonian operators namely expressed
with the gradient, divergence and rotationl, which need to get the corresponding
Euler equations, in order to improve the work efficiency. For this purpose using the
variational transform technology of the scalar and vector and the global variation
technology of the variable, by directly performing the variation to the gradient,
divergence and rotation of the vector, prove the theorem of variational problem of
such functional, give the Euler equations and natural boundary conditions expressed
by the scalar, vector, modulus of vector and Hamiltonian operators, give the corollary
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_10
609
Variational Problems of Functionals
with Vector, Tensor and Hamiltonian
Operators
The variational problems of functionals with the scalar, vector, modulus of vector,
tensor, trace of tensor, transposed tensor, Hamiltonian operator and Hamiltonian
operator string are discussed in this chapter, which include the functionals of any
complex form of vectors, nth order tensor and its gradient, divergence and rotation,
for this purpose, a series of new concepts, new theories and new ways of thinking are
advanced, in order to build the science building of the variational operation of the
above mentioned functionals, derive the corresponding Euler equations and natural
boundary conditions. The general method of variational problem of the functional
with Hamiltonian operators has been found, this is taking the divergence operation
of the vector as the starting point. With the methods advanced in this chapter it
can establish an infinite number of Euler equations of the functionals containing the
scalar, vector, tensor and Hamiltonian operators and obtain the corresponding natural
boundary conditions.
According to the need of various variational operations of the functionals with
the tensor, first to introduce the basic properties of the parallel type and series type
two kinds of inner product operations of the two nth order tensors, by the way to give
the variational formula of the nth order transposed tensor. On this basis, advance the
fundamental lemma of the functional variation, which lays a theoretical foundation
for the operation of variational problem of the functional with the vector, tensor and
Hamiltonian operators.
With the development of science and technology and the progress of the society,
more and more research fields involve the variational problems of the functional with
the scalar, vector, modulus of vector and Hamiltonian operators namely expressed
with the gradient, divergence and rotationl, which need to get the corresponding
Euler equations, in order to improve the work efficiency. For this purpose using the
variational transform technology of the scalar and vector and the global variation
technology of the variable, by directly performing the variation to the gradient,
divergence and rotation of the vector, prove the theorem of variational problem of
such functional, give the Euler equations and natural boundary conditions expressed
by the scalar, vector, modulus of vector and Hamiltonian operators, give the corollary
© Beijing Institute of Technology Press and Springer Nature Singapore Pte Ltd. 2021
D. Lao and S. Zhao, Fundamental Theories and Their Applications of the Calculus
of Variations, https://doi.org/10.1007/978-981-15-6070-5_10
609