384
7 Variational Principles
principle. Briefly, a scientific law presented in the form of variation is called the
variation(al) principle, in other words a scientific laws is boiled down to the variational problem of a functional is called the variation(al) principle. Historically,
the variational principle in produce had been touched with religion and metaphysics.
The most fundamental characteristics of variational principle is that it has nothing to
do with the choice of the coordinate system. The essence of variational principle is
that the first variation of the functional is zero, in other words, that the first variation
of the functional is equal to zero is the most fundamental variational principle. Since
the variational principle is rigorous, concise and beautiful in the mathematical form,
common, deep and rich in the physical content, can bottom the general theory, hence
the variational principle is considered to be the highest form of the scientific laws. The
variational principle reflects the unity of the objective world. The variational principle
helps people to find more new things, to explore more new territories. Sometimes the
variational principle also contains the meaning of the variational method, such as Weizang Chien’s masterpiece 《变 变分 分法 法及 及有 有限 限元 元》 , its English translation is “Variational
principles and finite element Methods”. Like Chap. 5, if the Lagrange multiplier
is introduced, the variational problem of conditional extremum is transformed into
the variational problem of unconditional extremum, then the corresponding variational principle obtained from which is called the generalized variational principle.
The variational principle that all the constraint conditions are removed is called the
completely generalized variational principle, it is called the generalized variational principle or the modified variational principle for short. The variational
principle that the partial constraint conditions are removed is called the incompletely
generalized variational principle. Due to the variational problem corresponding to
the boundary value problem of the second order differential equation, Usually the
integrand in which about unknown function and its derivative are secondary, namely
the extremal problem of quadratic functional, such functional in physics often represents energy, the functional represented by energy is called the energy functional
or functional of energy, thus customarily the method of solution transforming the
boundary value problem of the second order differential equation into the extremal
problem of a quadratic functional is called the energy method, in other words, the
variational method based on the extremal problem of energy functional is called the
energy method, the corresponding quadratic functional is called the energy integral
of the differential equation.
This chapter discusses how to convert the differential equations into the functionals equivalent to them, that is the variational principle. Of course, this involves
some concepts of function of real variable and functional analysis, this chapter
introduces first.
7.1 Sets and Mappings
In a certain range all the objects which has some properties or satisfies certain conditions is called a set. Each object in a set is called the element or member. The
elements in a set should be different. The element in a set is called a point. Usually
7 Variational Principles
principle. Briefly, a scientific law presented in the form of variation is called the
variation(al) principle, in other words a scientific laws is boiled down to the variational problem of a functional is called the variation(al) principle. Historically,
the variational principle in produce had been touched with religion and metaphysics.
The most fundamental characteristics of variational principle is that it has nothing to
do with the choice of the coordinate system. The essence of variational principle is
that the first variation of the functional is zero, in other words, that the first variation
of the functional is equal to zero is the most fundamental variational principle. Since
the variational principle is rigorous, concise and beautiful in the mathematical form,
common, deep and rich in the physical content, can bottom the general theory, hence
the variational principle is considered to be the highest form of the scientific laws. The
variational principle reflects the unity of the objective world. The variational principle
helps people to find more new things, to explore more new territories. Sometimes the
variational principle also contains the meaning of the variational method, such as Weizang Chien’s masterpiece 《变 变分 分法 法及 及有 有限 限元 元》 , its English translation is “Variational
principles and finite element Methods”. Like Chap. 5, if the Lagrange multiplier
is introduced, the variational problem of conditional extremum is transformed into
the variational problem of unconditional extremum, then the corresponding variational principle obtained from which is called the generalized variational principle.
The variational principle that all the constraint conditions are removed is called the
completely generalized variational principle, it is called the generalized variational principle or the modified variational principle for short. The variational
principle that the partial constraint conditions are removed is called the incompletely
generalized variational principle. Due to the variational problem corresponding to
the boundary value problem of the second order differential equation, Usually the
integrand in which about unknown function and its derivative are secondary, namely
the extremal problem of quadratic functional, such functional in physics often represents energy, the functional represented by energy is called the energy functional
or functional of energy, thus customarily the method of solution transforming the
boundary value problem of the second order differential equation into the extremal
problem of a quadratic functional is called the energy method, in other words, the
variational method based on the extremal problem of energy functional is called the
energy method, the corresponding quadratic functional is called the energy integral
of the differential equation.
This chapter discusses how to convert the differential equations into the functionals equivalent to them, that is the variational principle. Of course, this involves
some concepts of function of real variable and functional analysis, this chapter
introduces first.
7.1 Sets and Mappings
In a certain range all the objects which has some properties or satisfies certain conditions is called a set. Each object in a set is called the element or member. The
elements in a set should be different. The element in a set is called a point. Usually