Calculus of variations

The calculus of variations is a division of mathematics, which was developed in the mid- 18th century, in particular by Leonhard Euler and Joseph -Louis Lagrange.

The central element of the calculus of variations is the Euler -Lagrange equation

Which is the Lagrange equation from classical mechanics for now.

Basics

The calculus of variations deals with real functions of functions, which are also called functionals. Such functionals can be as integrals over an unknown function and its derivatives. Here we are interested for stationary functions, ie those for which the functional becomes a maximum, a minimum or a saddle point, it is called extremal. Some classical problems can be elegantly formulated in terms of functionals.

The Schlüsseltheorem the calculus of variations is the Euler -Lagrange equation, more precisely, " Euler - Lagrange differential equation ." This describes the Stationäritätsbedingung a functional. As with the task of determining the maxima and minima of a function, it is derived from the analysis of small changes to the solution adopted. Euler - Lagrange differential equation is only a necessary condition. Other necessary conditions for the existence of an extremal Adrien -Marie Legendre delivered and Alfred Clebsch and Carl Gustav Jacob Jacobi. A sufficient but not necessary condition comes from Karl Weierstrass.

The methods of the calculus of variations appear in the Hilbert space techniques of Morse theory and symplectic geometry. The term variation is used for all the problems of extremal features. Geodesy and differential geometry are areas of mathematics in which variations play a role. Especially the problem of minimal surfaces, which occur at about soap bubbles, has been working a lot.

Areas of application

The calculus of variations is the mathematical basis of all physical extremal and therefore especially in theoretical physics important, such as in the Lagrangian formalism of classical mechanics or the orbit determination, in quantum mechanics, in application of the principle of least action and in statistical physics in the context of density functional theory. In mathematics, the calculus of variations has been used for example in the Riemannian treatment of the Dirichlet principle for harmonic functions. Also in the control and regulation theory is the calculus of variations apply when it comes to the determination of optimal controllers.

A typical application example is the Brachistochronenproblem: What kind of curve in a gravitational field of a point A to a point B, which, however, is not below directly under A, an object requires the least time to traverse the curve? All curves of A and B, minimizing the expression that describes the time of passing through the curve. This expression is an integral, which contains the unknown, unknown function which describes the curve from A to B, and their derivatives.

An aid in analysis of real functions in one real variable

In the following, an important technique of variational calculus is demonstrated, in which a necessary statement for a local minimizer of a real function with only one real variable in a necessary proposition is transmitted for a local minimizer of a functional. This statement can then often be used for setting up descriptive equations for stationary functions of a functional.

Be given a functional on a function space ( must be at least a topological space to be). The functional have a local minimum at the location.

By following these simple trick takes the place of the " difficult to handle " functionals is a real function that depends only on a real parameter " is easier to treat and accordingly ".

With a be an arbitrary continuously parametrized by the real parameter family of functions. In this case, the function is (i.e., on ) is exactly equal to the fixed feature. In addition, whether by the equation

Defined function differentiable at the point.

The continuous function then takes the place of a local minimum, a local minimum as is.

From the analysis of real-valued functions in one real variable is known to be true. Applied to the functional means

When setting up the desired equations for stationary functions is then exploited that the above equation for any ( " benign " ) family must apply.

This will be demonstrated in the next section based on the Euler equation.

Euler-Lagrange equation; Variation of dissipation; Another necessary and sufficient conditions

Given two points in time and with a doubly continuously differentiable in all arguments function, the Lagrangian

For example, in the Lagrangian of the free relativistic particle with mass and

The territory of the Cartesian product of and the interior of the unit sphere.

As a function space is the set of all twice continuously differentiable functions

Chosen taking at the initial time point and end time point, the fixed predetermined locations or:

And whose values ​​are the values ​​together with its derivative,

With the Lagrangian now is the functional, the effect by

Defined. Sought is the function that minimizes the effect.

According to the technique presented in the previous section, we examine to all differentiable one-parameter families who go for the stationary operation of the functional ( it is so ). Used the equation derived in the last section

Pull-in of differentiation with respect to the parameters in the integral gives the chain rule

It stand for the derivatives with respect to the second and third argument and for the partial derivative with respect to the parameter.

It will prove to be beneficial later when in the second integral place as it is in the first integral. This is achieved by integration by parts:

At the points and apply regardless of the conditions and. Deriving this provides both Constant. Therefore, the term vanishes and, after combining the integrals and factoring out of the equation

And with

Except for the initial time and the end time is not restricted. Thus, the time functions are up to the conditions arbitrary twice continuously differentiable functions of time. The last equation can therefore only be met for all permissible if the factor throughout the integration interval is equal to zero ( which is a little more detail explained in the comments ). This gives the Euler -Lagrange equation for the stationary function

Which must be met for all.

The specified, the disappearance to be brought size is also referred to as the Euler derivative of the Lagrangian

Especially in physics books, the derivative is designated as a variation. Then, the variation of the variation of the effect

Is as a linear form in the variations of the arguments, their coefficients of variation are called derivation of the functional it is the case under consideration, the Euler derivative of the Lagrangian

Comments

In deriving the Euler-Lagrange equation is used that is a continuous function, to all the at least twice continuously differentiable functions in integration over

Null results, must be identically equal to zero.

Which is easy to see when it is considered that it for example with

Is a twice continuously differentiable function that is positive in an environment of a randomly chosen time point and zero otherwise. There would be a point at which the feature is greater than or less than zero, it would be because of the continuity even in a whole area of the body is greater or less than zero. With the function just defined, however, then the integral in contradiction to the requirement to also greater or less zero. The assumption that at one point would be equal to zero, so is wrong. The function is therefore really identical zero.

If the function space an affine space, so the family is often defined in the literature as the sum of an arbitrary function of time, which must satisfy the condition. The derivative is then just the gateaux derivative of the functional position in the direction. The version presented here seems to the author a bit cheaper, if the set of functions affine space no longer (if it is for example limited by a non-linear constraint, see eg Gaussian principle of least constraint ). It is shown in detail and is based on the definition of tangent vectors to manifolds (see also ).