Functional and variational basis

Basic Concept Functional

A functional is an expression of a function, and the value depends on the function in the expression, which is the function of the function.

1) In addition to the variable x, the functional can also contain other independent variables;

2) In addition to the function y (x), the functional can also contain many other functions (dependent variables) which are functions of the independent variables mentioned above;

3) In a functional function, in addition to the first derivative, it can also contain higher order derivatives.

Variational method

The classical variational problem is the optimal solution to a problem, and the process of solving it is "optimization".

The method and process of solving the classical variational problem is the method and process of the function seeking extremum.

The method of studying functional extremum is called variational method, and the approximate method of studying functional extremum is called variational method.

First-order variational

function f Pairs of variables x, y and y ' two times can be micro;

The number of functional I at two points depends on the path selected between two points, which is the function y (x).

The function y (x) is set so that the functional I reaches the extremum and its adjacent path is .

Y (x) is called the "extremum curve" or "extremum function", called the "variable path".

is a micro-function, A is a trace of a parameter variable.

When a=0, the necessary conditions for extreme values are:

Notice that the

Point at both ends = 0

Because for any function

so that is Euler-Lagrange equation

Variational operations

Introduce " operator", define

operator means that when the independent variable x is a fixed value, any minor change of the dependent variable function Y.

The use of F is written along a variable curve:

At any x, expand F to the Taylor series about Y and y ':

Total variation of F:

First-order variational points:

I take the extremum condition:

has multiple dependent variables:

Contains derivatives number:

(Euler-poisson equation )

Functional and variational basis