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Generating function of canonical transformation

Lecture



In Hamiltonian mechanics, canonical transformation (also contact transformation ) is a transformation of canonical variables and a Hamiltonian that does not change the general form of the Hamilton equations for any Hamiltonian system. Canonical transformations can also be introduced in the quantum case as non-changing form of Heisenberg equations. They allow us to reduce a problem with a certain Hamiltonian to a problem with a simpler Hamiltonian in both the classical and quantum cases. Canonical transformations form a group.

Definition

Transformations

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation where Generating function of canonical transformation - the number of degrees of freedom

Generating function of canonical transformation

are called canonical if this transformation translates the Hamilton equations with the Hamilton function Generating function of canonical transformation :

Generating function of canonical transformation

Generating function of canonical transformation

Hamilton equations with Hamilton function Generating function of canonical transformation :

Generating function of canonical transformation

Generating function of canonical transformation

Variables Generating function of canonical transformation and Generating function of canonical transformation are called new coordinates and momenta, respectively, and Generating function of canonical transformation and Generating function of canonical transformation - old coordinates and pulses.

Performance Functions

From the invariance of the Poincaré – Cartan integral and the Lee Hua-jung theorem on its uniqueness, we can obtain:

Generating function of canonical transformation

where is the constant Generating function of canonical transformation called the valence of the canonical transformation, Generating function of canonical transformation - total differential of some function Generating function of canonical transformation (it is assumed that Generating function of canonical transformation and Generating function of canonical transformation also expressed in terms of old variables). It is called the generating function of the canonical transformation. Canonical transformations are one-to-one defined by the generating function and valence.

Canonical transformations for which Generating function of canonical transformation called univalent . Since for a given generating function different Generating function of canonical transformation if the expressions for the new coordinates are changed through the old ones, and also for the Hamiltonian only by a constant, then only univalent canonical transformations are often considered.

The generating function can often be expressed not through old coordinates and impulses, but through any two of the four variables. Generating function of canonical transformation , and the choice is independent for each Generating function of canonical transformation . It is convenient to express it so that for everyone Generating function of canonical transformation one variable was new and the other was old. There is a lemma stating that this can always be done. Differential function Generating function of canonical transformation has the explicit form of the total differential in the case when it is expressed through the old and new coordinates Generating function of canonical transformation . When using other coordinate pairs, it is convenient to go to functions whose differential will have the explicit form of the total differential for the corresponding variables. For this you need to do the Legendre transform of the original function Generating function of canonical transformation . The resulting functions are called the generating functions of the canonical transformation in the corresponding coordinates. In the case when the choice of coordinates is the same for all Generating function of canonical transformation There are four possible choices of variables, the corresponding functions are usually denoted by numbers:

Generating function of canonical transformation

where for simplicity vectors of old velocities and impulses are introduced Generating function of canonical transformation , Generating function of canonical transformation , similarly for new speeds and impulses. Such generating functions are referred to as generating functions of the 1st, 2nd, 3rd, or 4th type, respectively.

Type 1 generating function

Let be Generating function of canonical transformation - arbitrary non-degenerate function of old coordinates, new coordinates and time:

Generating function of canonical transformation

besides, some number is given Generating function of canonical transformation then a couple Generating function of canonical transformation defines a canonical transformation by rule

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

Relation to the original generating function:

Generating function of canonical transformation

The canonical transformation can be obtained with the help of such a function if the Jacobian is not equal to zero:

Generating function of canonical transformation

Canonical transformations supplemented with this condition are called free .

Generating function of the 2nd type

Let be Generating function of canonical transformation - arbitrary non-degenerate function of old coordinates, new coordinates and time:

Generating function of canonical transformation

besides, some number is given Generating function of canonical transformation then a couple Generating function of canonical transformation defines a canonical transformation by rule

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

Relation to the original generating function:

Generating function of canonical transformation

The canonical transformation can be obtained with the help of such a function if the Jacobian is not equal to zero:

Generating function of canonical transformation

Type 3 generating function

Let be Generating function of canonical transformation - arbitrary non-degenerate function of old coordinates, new coordinates and time:

Generating function of canonical transformation

besides, some number is given Generating function of canonical transformation then a couple Generating function of canonical transformation defines a canonical transformation by rule

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

Relation to the original generating function:

Generating function of canonical transformation

A canonical transformation can be obtained using such a function if the Jacobian is not equal to zero:

Generating function of canonical transformation

Type 4 generating function

Let be Generating function of canonical transformation - arbitrary non-degenerate function of old coordinates, new coordinates and time:

Generating function of canonical transformation

besides, some number is given Generating function of canonical transformation then a couple Generating function of canonical transformation defines a canonical transformation by rule

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

Relation to the original generating function:

Generating function of canonical transformation

The canonical transformation can be obtained with the help of such a function if the Jacobian is not equal to zero:

Generating function of canonical transformation

Examples

1. Identical transformation

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

can be obtained by:

Generating function of canonical transformation

2. If you specify

Generating function of canonical transformation

then the resulting transformation will be:

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

Thus, the separation of canonical variables into coordinates and impulses from a mathematical point of view is conditional.

3. Inversion conversion

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

can be obtained by:

Generating function of canonical transformation

4. Point transformations (transformations in which new coordinates are expressed only through old coordinates and time, but not old impulses.)

They can always be specified using:

Generating function of canonical transformation

then

Generating function of canonical transformation

In particular, if

Generating function of canonical transformation

Where Generating function of canonical transformation - orthogonal matrix:

Generating function of canonical transformation

that

Generating function of canonical transformation

Generating function of canonical transformation

The function also leads to point transformations:

Generating function of canonical transformation

then

Generating function of canonical transformation

In particular, the function

Generating function of canonical transformation

sets the transition from Cartesian to cylindrical coordinates.

5. Linear variable transformations Generating function of canonical transformation systems with one degree of freedom:

Generating function of canonical transformation

Generating function of canonical transformation

is a univalent canonical transformation with

Generating function of canonical transformation

generating function:

Generating function of canonical transformation

Such transformations form a special linear group. Generating function of canonical transformation .

Action as a generating function

Action expressed as a function of endpoint coordinates and momenta

Generating function of canonical transformation

specifies the canonical transformation of the Hamiltonian system.

Poisson and Lagrange brackets

The necessary and sufficient condition for the canonicity of transformations can be written using the Poisson brackets:

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

In addition, a necessary and sufficient condition for the canonicity of the transformation is the performance for arbitrary functions Generating function of canonical transformation and Generating function of canonical transformation conditions:

Generating function of canonical transformation

where under Generating function of canonical transformation and Generating function of canonical transformation Poisson brackets are understood by the old and new coordinates, respectively.

In the case of univalent canonical transformations:

Generating function of canonical transformation

and say that Poisson brackets are invariant with respect to such transformations. Sometimes canonical transformations are defined this way (in this case, only univalent transformations are considered canonical transformations).

Similarly, the necessary and sufficient condition for the canonicity of transformations can be written using Lagrange brackets:

Generating function of canonical transformation

Generating function of canonical transformation

Generating function of canonical transformation

created: 2014-12-31
updated: 2021-12-26
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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis