Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations and Liouville's theorem.
Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into
where are the new co‑ordinates, grouped in canonical conjugate pairs of momenta and corresponding positions for with being the number of degrees of freedom in both co‑ordinate systems.
Therefore, coordinate transformations are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations.
Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles, exterior derivatives and symplectic manifolds.

Notation

Boldface variables such as represent a list of generalized coordinates that need not transform like a vector under rotation and similarly represents the corresponding generalized momentum, e.g.,
A dot over a variable or list signifies the time derivative, e.g.,
and the equalities are read to be satisfied for all coordinates, for example:
The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g.,
The dot product maps the two coordinate lists into one variable representing a single numerical value. The coordinates after transformation are similarly labelled with for transformed generalized coordinates and for transformed generalized momentum.

Conditions for restricted canonical transformation

Restricted canonical transformations are coordinate transformations where transformed coordinates and do not have explicit time dependence, i.e., and. The functional form of Hamilton's equations is
In general, a transformation does not preserve the form of Hamilton's equations but in the absence of time dependence in transformation, some simplifications are possible. Following the formal definition for a canonical transformation, it can be shown that for this type of transformation, the new Hamiltonian can be expressed as:
where it differs by a partial time derivative of a function known as a generator, which reduces to being only a function of time for restricted canonical transformations.
In addition to leaving the form of the Hamiltonian unchanged, it is also permits the use of the unchanged Hamiltonian in the Hamilton's equations of motion due to the above form as:
Although canonical transformations refers to a more general set of transformations of phase space corresponding with less permissive transformations of the Hamiltonian, it provides simpler conditions to obtain results that can be further generalized. All of the following conditions, with the exception of bilinear invariance condition, can be generalized for canonical transformations, including time dependence.

Indirect conditions

Since restricted transformations have no explicit time dependence, the time derivative of a new generalized coordinate is

where is the Poisson bracket.
Similarly for the identity for the conjugate momentum, P_m using the form of the "Kamiltonian" it follows that:
Due to the form of the Hamiltonian equations of motion,
if the transformation is canonical, the two derived results must be equal, resulting in the equations:
The analogous argument for the generalized momenta P_m leads to two other sets of equations:
These are the indirect conditions to check whether a given transformation is canonical.

Symplectic condition

Sometimes the Hamiltonian relations are represented as:
Where
and. Similarly, let.
From the relation of partial derivatives, converting the relation in terms of partial derivatives with new variables gives where. Similarly for,
Due to form of the Hamiltonian equations for,
where can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:
The left hand side of the above is called the Poisson matrix of, denoted as. Similarly, a Lagrange matrix of can be constructed as. It can be shown that the symplectic condition is also equivalent to by using the property. The set of all matrices which satisfy symplectic conditions form a symplectic group. The symplectic conditions are equivalent with indirect conditions as they both lead to the equation, which is used in both of the derivations.

Invariance of the Poisson bracket

The Poisson bracket which is defined as:can be represented in matrix form as:
Hence using partial derivative relations and symplectic condition gives:
The symplectic condition can also be recovered by taking and which shows that. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that, which is also the result of explicitly calculating the matrix element by expanding it.

Invariance of the Lagrange bracket

The Lagrange bracket which is defined as:
can be represented in matrix form as:
Using similar derivation, gives:
The symplectic condition can also be recovered by taking and which shows that. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that, which is also the result of explicitly calculating the matrix element by expanding it.

Bilinear invariance conditions

These set of conditions only apply to restricted canonical transformations or canonical transformations that are independent of time variable.
Consider arbitrary variations of two kinds, in a single pair of generalized coordinate and the corresponding momentum:
The area of the infinitesimal parallelogram is given by:
It follows from the symplectic condition that the infinitesimal area is conserved under canonical transformation:
Note that the new coordinates need not be completely oriented in one coordinate momentum plane.
Hence, the condition is more generally stated as an invariance of the form under canonical transformation, expanded as:
If the above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met.
The form of the equation, is also known as a symplectic product of the vectors and and the bilinear invariance condition can be stated as a local conservation of the symplectic product.

Liouville's theorem

The indirect conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e.,
By calculus, the latter integral must equal the former times the determinant of Jacobian
Where
Exploiting the "division" property of Jacobians yields
Eliminating the repeated variables gives
Application of the indirect conditions above yields.

Generating function approach

To guarantee a valid transformation between and, we may resort to a direct generating function approach. Both sets of variables must obey Hamilton's principle. That is the action integral over the Lagrangians and, obtained from the respective Hamiltonian via an "inverse" Legendre transformation, must be stationary in both cases :
One way for both variational integral equalities to be satisfied is to have
Lagrangians are not unique: one can always multiply by a constant and add a total time derivative and yield the same equations of motion. In general, the scaling factor is set equal to one; canonical transformations for which are called extended canonical transformations. is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.
Here is a generating function of one old canonical coordinate, one new canonical coordinate and the time. Thus, there are four basic types of generating functions, depending on the choice of variables. As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation is guaranteed to be canonical.
The various generating functions and its properties tabulated below is discussed in detail:

Type 1 generating function

The type 1 generating function depends only on the old and new generalized coordinates. To derive the implicit transformation, we expand the defining equation above
Since the new and old coordinates are each independent, the following equations must hold
These equations define the transformation as follows: The first set of equations define relations between the new generalized coordinates and the old canonical coordinates. Ideally, one can invert these relations to obtain formulae for each as a function of the old canonical coordinates. Substitution of these formulae for the coordinates into the second set of equations yields analogous formulae for the new generalized momenta in terms of the old canonical coordinates. We then invert both sets of formulae to obtain the old canonical coordinates as functions of the new canonical coordinates. Substitution of the inverted formulae into the final equation yields a formula for as a function of the new canonical coordinates.
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let. This results in swapping the generalized coordinates for the momenta and vice versa
and. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.