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 (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

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 being the number of degrees of freedom in both coordinate systems.

Therefore, coordinate transformations (also called point 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 (many textbooks consider only this type).

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

A dot over a variable or list signifies the time derivative

The dot product

The dot product (also known as an "inner 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 . The functional form of Hamilton's equations is

In general, a transformation does not preserve the form of Hamilton's equations but in the case of a time independent 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 (sometimes called the Kamiltonian) can be expressed as:

where it differs from the Hamiltonian 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, the Kamiltonian also permits the use of the unchanged Hamiltonian in Hamilton's equations of motion due to the above form as:

Although the class of canonical transformations is larger than that which can be obtained with this restricted formulation, the restricted version provides a good starting point to obtain results that can be further generalized. All of the following conditions, with the exception of bilinear invariance condition, can be generalized for the full class of canonical transformations, including time dependence.

Indirect conditions

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

where is the Poisson bracket.

Similarly for the identity for the conjugate momentum, Pm 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 Pm 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:

.

,

,

can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:

. which satisfy symplectic conditions form a symplectic group, which is used in both of the derivations.

Invariance of the Poisson bracket

The Poisson bracketcan be represented in matrix form as:

Hence using partial derivative relations and symplectic condition gives:

, 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:

, 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:

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.

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. 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

Exploiting the "division" property of Jacobians

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, obtained from the respective Hamiltonian via an "inverse" Legendre transformation, must be stationary in both cases (so that one can use the Euler–Lagrange equations to arrive at Hamiltonian equations of motion of the designated form; as it is shown for example here):

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 (as discussed on Wikibooks). 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 ( or ), one new canonical coordinate ( or ) and (possibly) the time . Thus, there are four basic types of generating functions (although mixtures of these four types can exist), 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

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 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 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 yields a formula for as a function of the new canonical coordinates .

. 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.

Type 2 generating function

depends only on the old generalized coordinates terms represent a Legendre transformation to change the right-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above

Since the old coordinates and new momenta are each independent, the following equations must hold

These equations define the transformation as follows: The first set of define relations between the new generalized momenta 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 yields analogous formulae for the new generalized coordinates 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 yields a formula for as a function of the new canonical coordinates .

where is a set of .

Type 3 generating function

terms represent a Legendre transformation

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 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 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 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.

Type 4 generating function

terms represent a Legendre transformation to change both sides of the equation below. 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 define relations between the new generalized momenta 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 yields analogous formulae for the new generalized coordinates 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 yields a formula for as a function of the new canonical coordinates .

Limitations on the four types of generating functions

is non-singular using the inverse function theorem, and can be restated as the following relation.

, being non-singular. matrix. Hence, the canonical transformations resulting from these four generating functions alone are not completely general.

Generalized use of generating functions

In other words, since and are each are restricted to be non singular, ensuring that the generating function is a function of independent variables. However, as a feature of canonical transformations, it is always possible to choose such independent functions from sets or , to form a generating function representation of canonical transformations, including the time variable. Hence, it can be proven that every finite canonical transformation can be given as a closed but implicit form that is a variant of the given four simple forms.

Canonical transformation conditions

Canonical transformation relations

:

.

.

Similarly:

.

in this context.

The canonical transformation relations can now be restated to include time dependence:

, if and can be taken. The analysis of restricted canonical transformations is hence consistent with this generalization.

Symplectic condition

, in Hamiltonian's equations gives:

:

or:

, from which the indirect conditions can be recovered. Thus, symplectic conditions and indirect conditions can be said to be equivalent in the context of using generating functions.

Invariance of the Poisson and Lagrange brackets

are obtained which imply the invariance of Poisson and Lagrange brackets.

Extended canonical transformation

Canonical transformation relations

By solving for:

case.

from known solutions, since it retains the form of Hamilton's equations which satisfies the condition).

, extended canonical transformation partial differential relations are obtained as:

Symplectic condition

Following the same steps to derive the symplectic conditions, as:

and

instead gives:

.

Poisson and Lagrange brackets

The Poisson brackets are changed as follows:

whereas, the Lagrange brackets are changed as:

.

Infinitesimal canonical transformation

, as follows:

, the corresponding transformations are called as infinitesimal canonical transformations which are also known as differential canonical transformations.

Explicit construction

Consider the following generating function:

to an infinitesimal value.

From the conditions of generators of second type:

, gives:

Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition. is very significant in infinitesimal canonical transformations and is referred to as the generator of infinitesimal canonical transformation.

Active and passive transformations

In the active view of transformations, the coordinate system is changed without the physical system changing, whereas in the passive view of transformation, the coordinate system is retained and the physical system is said to undergo transformations.

Active view of transformation

Thus, using the relations from infinitesimal canonical transformations, the change in the system states under active view of the canonical transformation is said to be:

in matrix form.

, it changes under active view of the transformation according to:

Passive view of transformation

Considering the change of Hamiltonians in the passive view

. Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.

Generators of dynamical symmetry transformations

Consider the transformation where the change of coordinates also depends on the generalized velocities.

If the above is a dynamical symmetry, then the lagrangian changes by:

and the new Lagrangian is said to be dynamically equivalent to the old Lagrangian as it ensures the resultant equations of motion being the same. The change in generalized velocity and momentum term can be derived as:

Generator of transformation

Using the change in Lagrangian property of a dynamical symmetry:

using which, it can be shown that

which is a infinitesimal point transformation. Note that in the point transformation condition, the quantity generates the transformation regardless of if the Euler Lagrange equations are satisfied and since they do not depend on the dynamics of the problem are said to be a purely kinematic relation.

of Lorentz invariance generates Lorentz transformation of fields.

Noether Invariant

is a constant of motion. Hence, the derived Noether invariant also generates the same symmetry transformation as shown previously.

Examples of ICT

Time evolution

. Hence if the Hamiltonian is time translation invariant, i.e. does not have explicit time dependence, its value is conserved for the motion.

Translation

. Hence, the canonical momentum generates a shift in the corresponding generalized coordinate and if the Hamiltonian is invariant of translation, the momentum is a constant of motion.

Rotation

Consider an orthogonal system for an N-particle system:

, then the change in the coordinates is given for x by:

and similarly for y:

.

in its first order approximation. Hence, repeated application of the infinitesimal canonical transformation generates a rotation of system of particles about the z axis. If the Hamiltonian is invariant under rotation about the z axis, the generator, the component of angular momentum along the axis of rotation, is an invariant of motion.

One parameter subgroup of Canonical transformations

to take continuous range of values in:

.

.

The condition on the transformations of the one parameter subgroup kind can be expressed equivalently as a differential equation:

.

.

This can be used to express the differential equation as:

where the linear differential operator.

Active view of transformation

Upon iteratively solving the differential equation, the solution of the differential equation follows as:

we get similarly

Passive view of transformation

can be expressed as upto first order as:

which indicates that the function transforms oppositely to that due to the coordinates to preserve well defined mapping from a physical point in phase space to its scalar values. It is also possible that functions transform without needing to preserve its values on the same physical states in phase space. Such as, for example, the Hamiltonian whose explicit dependence on the canonical transformation can be different from the above form, restated from its previous derivation as

which is similar to previous relation but also accounts for any explicit time dependence of the generator. Hence, if the Hamiltonian is invariant in passive view for infinitesimal canonical transformations, its generator is a constant of motion.

Motion as canonical transformation

, then Hamilton's principle should always satisfy Hamilton's principle, regardless of the endpoints.

Examples

• .

• is a rotation matrix independently, so these are not the same as a physical rotation of an orthogonal spatial coordinate system.

• which is symplectic.

Modern mathematical description

In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates is written here as a superscript), not as a subscript). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.

History

The first major application of the canonical transformation was in 1846, by Charles Delaunay, in the study of the Earth-Moon-Sun system. This work resulted in the publication of a pair of large volumes as Mémoires by the French Academy of Sciences, in 1860 and 1867.

See also

• Symplectomorphism

• Hamilton–Jacobi equation

• Liouville's theorem (Hamiltonian)

• Mathieu transformation

• Linear canonical transformation

Notes

References