Chapter 7
Variational Principles
The problems discussed in the previous chapters are how to transform the extremal
problems of solving the functionals into the solution problems of the Euler equation or Ostrogradsky equation, in most cases, they respectively correspond to the
boundary value problems of ordinary differential equation and partial differential
equation. But in many cases, to solve the boundary value problem of differential
equation is also very difficult. That the variational problems are boiled down to the
boundary value problem of differential equation sometimes conversely complicate
the problem. Now consider the opposite problem, if a boundary value problem of
differential equation has been known, then can it be transformed into the extremal
problem of a functional and solved, with approximate method? This is so called
the variational method of solution of the variational problem for differential equation, this is a very important and difficult problem. In response to the problem,
American mathematician Friedrich once gave proof: For a positive definite operator
equation, there must be a minimum problem of the functional equivalent to it. In
other words, if the solution of minimum problem of the functional can be found,
the corresponding solution of operator equation can also be found. The boundary
value problem of differential equation is transformed into the functional equivalent
to it, namely the Euler equation of the functional is the given differential equation,
this problem is called the contrary problem of variational problem or contrary
variational problem, also called the inverse problem of variational problem or
inverse variational problem. This functional often expresses energy in physics, it is
called the energy integral of the original differential equation. The solving method
the theory that the boundary value problem of differential equation is transformed
into the equivalent extremal problem of the functional is called the variation(al)
principle, variational method or variational approach. The definition of the variational principle is also given in Chap. 2, integrating the two definitions, the method
the theory that the boundary value problem of differential equation is transformed
into the equivalent extremal problem of the functional equalent to it and solving
the extremal problem of the functional namely the variational problem are both
called the variation(al) principle. There are many statements about the variational
© 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_7
383
Variational Principles
The problems discussed in the previous chapters are how to transform the extremal
problems of solving the functionals into the solution problems of the Euler equation or Ostrogradsky equation, in most cases, they respectively correspond to the
boundary value problems of ordinary differential equation and partial differential
equation. But in many cases, to solve the boundary value problem of differential
equation is also very difficult. That the variational problems are boiled down to the
boundary value problem of differential equation sometimes conversely complicate
the problem. Now consider the opposite problem, if a boundary value problem of
differential equation has been known, then can it be transformed into the extremal
problem of a functional and solved, with approximate method? This is so called
the variational method of solution of the variational problem for differential equation, this is a very important and difficult problem. In response to the problem,
American mathematician Friedrich once gave proof: For a positive definite operator
equation, there must be a minimum problem of the functional equivalent to it. In
other words, if the solution of minimum problem of the functional can be found,
the corresponding solution of operator equation can also be found. The boundary
value problem of differential equation is transformed into the functional equivalent
to it, namely the Euler equation of the functional is the given differential equation,
this problem is called the contrary problem of variational problem or contrary
variational problem, also called the inverse problem of variational problem or
inverse variational problem. This functional often expresses energy in physics, it is
called the energy integral of the original differential equation. The solving method
the theory that the boundary value problem of differential equation is transformed
into the equivalent extremal problem of the functional is called the variation(al)
principle, variational method or variational approach. The definition of the variational principle is also given in Chap. 2, integrating the two definitions, the method
the theory that the boundary value problem of differential equation is transformed
into the equivalent extremal problem of the functional equalent to it and solving
the extremal problem of the functional namely the variational problem are both
called the variation(al) principle. There are many statements about the variational
© 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_7
383