Canonical transformation

In classical mechanics is called a coordinate transformation in the phase space as canonical if she lets the Hamiltonian equations invariant. The aim is to simplify the new Hamiltonian as possible to make even independent of one or more variables in the ideal case. Canonical transformations are the starting point for the Hamilton -Jacobi formalism and provide (as contact transformations ) a more powerful tool than point transformations that occur in the Lagrangian mechanics, as these emerge as a special case of the canonical transformations. Canonical transformation can be constructed from so-called generating functions.

Definition

Given a dynamic system with F degrees of freedom and the Hamiltonian function, which may depend on the co-ordinates and signals and of the time. The canonical equations ( Hamiltonian equations of motion ) are therefore:

A coordinate transformation means, and canonical, Hamiltonian when a new function exist, so that in the transformed coordinates of the canonical equations are satisfied:

Properties

Poisson brackets

The Poisson bracket of the functions f and g with respect to q and p by

Defined. If the transformation is canonical, the following applies:

In addition, one can find a general criterion using the Poisson brackets when a transformation is canonical, exactly when the fundamental Poisson brackets

Applies. It is the Kronecker delta.

Generating functions

Canonical transformations can be generating functions ( briefly also generatrix ) and have been constructed. Of these functions, there are four classes. For example,, the generating function of the old and new coordinates depend, therefore, other generating depend on other combinations of coordinates and impulses from, each transformed and untransformed variables must occur. The relationship between old and new coordinates can from the Hamiltonian variational

Be derived ( it is summation convention is used). The difference between the two expressions must also be zero and may therefore be only the total time derivative of a function, which variation disappears at the edges and in:

This leads to the relationship:

If a function of the old and new coordinates, so is the total time derivative

And you can read the partial derivatives of by or as well as from the previous line directly:

With the first two derivatives can be expressed as a function of the old coordinates of the new: and

By Legendre transformation of these equations is similar relationships between old and new coordinates can be derived. There are four classes, as the Legendre transformation for the transformations and conduct, there are 2 times 2 possibilities. As an example, we transform it to:

If on the left side

Obtained a function of, and is dependent, wherein the partial derivatives are as follows:

Overview of all possible generating functions:

For all four classes of generating functions is considered: The Hamiltonian in transformed coordinates differs from the untransformed Hamiltonian by a partial time derivative of the generating function so

Canonical transformations whose generating function does not depend explicitly on the time, called canonical transformations in the strict sense.

Special generating functions

If you choose the generating function it follows the identical transformation:

The transformation to the generating function, however, confuses the roles of coordinates and impulses:

In the following we consider a canonical transformation, the new Hamiltonian is identically zero:

Then it follows from the Hamiltonian equations of motion, that all variables are cyclically:

This leads to the relationship:

Or

That is equal to the effect of the system (more precisely, the indefinite integral effect ):

From results of the election, the differential equation

This equation is the Hamilton-Jacobi equation.

Symplectic structure

A second observation is that the canonical transformations Funktionalmatrizen, so

Form a symplectic group, ie have the following property:

With