Chapter 8
Direct Methods of Variational Problems
The final solution for all kinds of variational problems boils down to the boundary
value problem of solving the Euler equation. However, only in some special cases the
analytical solution of the Euler equation can be found, in most cases, the analytical
solution of the Euler equation can not be found. So it needs the other solution methods.
In August 1900, the 2nd international congress of mathematicians held in Paris,
Hilbert at the meeting made a lecture “mathematical problems”, put forward the
23 major mathematical problems, in which the last question was about the directly
solving problem of the calculus of variations, and the 19-th and 20-th questions also
involved the calculus of variations.
The direct methods of variational problems or direct methods in variational
problems refer to directly starting from the functional itself instead of solving the
Euler equation, find out the approximate expression of the extremal function that
makes the functional obtain extremum. Of course, sometimes the exact solution can
also be obtained. The preceding variational problems are obtained by the boundary
value problems of solving the Euler equation, this method of solution is called
the indirect method of variational problem or indirect method in variational
problem.
The approximate methods of solution for variational problem have the Euler finite
difference method, Ritz method, Kantorovich method, Galerkin method, least squares
method or method of least squares, Trefftz method, collocation method and partition
average method, etc. The Euler finite difference method, Ritz method, Kantorovich
method and Galerkin method are called the classical theory in the calculus of
variations or classical variational method. The former five kinds of approximate
methods and minimizing (maximizing) sequence are introduced in this chapter, and
the solving problem of the eigenvalue and eigenfunction of operator equation is
discussed.
© 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_8
453
Direct Methods of Variational Problems
The final solution for all kinds of variational problems boils down to the boundary
value problem of solving the Euler equation. However, only in some special cases the
analytical solution of the Euler equation can be found, in most cases, the analytical
solution of the Euler equation can not be found. So it needs the other solution methods.
In August 1900, the 2nd international congress of mathematicians held in Paris,
Hilbert at the meeting made a lecture “mathematical problems”, put forward the
23 major mathematical problems, in which the last question was about the directly
solving problem of the calculus of variations, and the 19-th and 20-th questions also
involved the calculus of variations.
The direct methods of variational problems or direct methods in variational
problems refer to directly starting from the functional itself instead of solving the
Euler equation, find out the approximate expression of the extremal function that
makes the functional obtain extremum. Of course, sometimes the exact solution can
also be obtained. The preceding variational problems are obtained by the boundary
value problems of solving the Euler equation, this method of solution is called
the indirect method of variational problem or indirect method in variational
problem.
The approximate methods of solution for variational problem have the Euler finite
difference method, Ritz method, Kantorovich method, Galerkin method, least squares
method or method of least squares, Trefftz method, collocation method and partition
average method, etc. The Euler finite difference method, Ritz method, Kantorovich
method and Galerkin method are called the classical theory in the calculus of
variations or classical variational method. The former five kinds of approximate
methods and minimizing (maximizing) sequence are introduced in this chapter, and
the solving problem of the eigenvalue and eigenfunction of operator equation is
discussed.
© 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_8
453