Legendre-Transformation

The Legendre transformation (after Adrien -Marie Legendre ) is one of the contact transformations and serves as an important mathematical method for transformation of variables.

The Legendre transformation of a variable

Is a function of a real variable which is referred to below. The Legendre transformation now performs this function in a function of the independent variable that is the derivative of the function after. A corresponding back-transformation is possible.

If so

Set, the unknown function is the equation

. meet This is

The reverse function.

It is noted that this equation of the function

Is achieved, as it is.

This function is called Legendre transform of.

For the inverse transformation

Set and then applies

Derivation

The aim of the Legendre transformation is the change of a function of a function of one variable to another variable for which holds

If you can write for the dependent function of

Then shall also apply to the dependent function of:

First we form the total differential of, we obtain

The comparison with and provides us

Thus, we conclude

After an integration therefore applies

The function is called the Legendre transform of. The sign of is not important for the definition, so we can or write. What is the sign you should choose depends on the physical meaning of.

Geometrical Meaning

Geometric can the facts as in the figure illustrate: The curve ( red) is used instead of the point set to specify from which it is made, be characterized by the set of all tangents envelop. This is what happens when the Legendre transformation. The transform, the tangent slope of each of which assigns the Y -intercept. So it is a description of the same curve - only via another parameter, namely instead.

For a function of several variables

The change in function of a function of an independent variable to another by means of a partial derivative of after is:

Herein, the geometric pitch in the x direction of the tangent plane to the function dar. Therefore, one speaks of contact transformation. The function is called the Legendre transform with respect to variable.

The Legendre transform can be derived as follows. The value of can alternatively as

Be written. Defines one now, one obtains for the Legendre transform

Usually will be selected and thus follows

For the latter definition the Legendre transform is the component of the intersection of the tangent plane to the plane. For functions in the plane is called the intercept ( see also linear equation ).

So practice, there is replacement of the independent variables by subtracting the product of old and new variable of the output function:

This is also evident when considering the total differential of the Legendre transform:

Areas of application

Use in physics is the Legendre transformation, especially in the (statistical) thermodynamics ( eg, the conversion of the fundamental equation or the transition between thermodynamic potentials under different boundary conditions) and the transition from Lagrangian to Hamiltonian mechanics ( Lagrangian to Hamilton function). In thermodynamics, using the lower sign convention ().

The Legendre transformation plays - like the touch of a total transformation - furthermore a role in the mechanics, the calculus of variations and in the theory of partial differential equations first order. In mechanics, one uses the upper sign convention ().

Examples

In mechanics is obtained by Legendre transformation of the Lagrangian, the Hamiltonian function and vice versa:

In thermodynamics, one can derive by Legendre transform from the fundamental equation of thermodynamics, the thermodynamic potentials. , For example, a transition of the internal energy U (depending on the entropy S) of the Helmholtz energy R (depending on the temperature T) instead. In the case of an ideal gas so the following applies:

Deriving notation used herein, derivative of U ( S, V, N) according to function S, where N and V are kept constant.

By analogy, is also a transition from a thermodynamic potential to another possible example of the enthalpy H of the Gibbs free energy G:

In the same manner, the other thermodynamic potential, wherein by a Legendre transformation is a generalized coordinate is always replaced by the conjugated thermodynamic force.