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).
For clarity, we restrict the presentation here to calculus and classical mechanics. Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article. (Canonical transformations are a special case of a symplectomorphism.) However, a brief introduction to the modern mathematical description is included at the end of this article.
A dot over a variable or list signifies the time derivative, e.g.,
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 (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value.
By definition, the transformed coordinates have analogous dynamics
where K(Q, P) is a new Hamiltonian (sometimes called the Kamiltonian[1]) that must be determined.
In general, a transformation (q, p, t) → (Q, P, t) does not preserve the form of Hamilton's equations. For time independent transformations between (q, p) and (Q, P) we may check if the transformation is restricted canonical, as follows. Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate Qm is
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 JacobianJ
To guarantee a valid transformation between (q, p, H) and (Q, P, K), 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 Lagrangian and respectively, obtained by the Hamiltonian via ("inverse") Legendre transformation, both must be stationary (so that one can use the Euler–Lagrange equations to arrive at equations of the above-mentioned and designated form; as it is shown for example here):
In general, the scaling factor λ is set equal to one; canonical transformations for which λ ≠ 1 are called extended canonical transformations. dG/dt 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 G is a generating function of one old canonical coordinate (q or p), one new canonical coordinate (Q or P) and (possibly) the time t. 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 (q, p) → (Q, P) is guaranteed to be canonical.
The type 1 generating function G1 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 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
define relations between the new generalized coordinatesQ and the old canonical coordinates(q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates(q, p). We then invert both sets of formulae to obtain the oldcanonical coordinates(q, p) as functions of the newcanonical coordinates(Q, P). Substitution of the inverted formulae into the final equation
The type 2 generating function G2 depends only on the old generalized coordinates and the new generalized momenta
where the 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 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
define relations between the new generalized momenta P and the old canonical coordinates(q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates(q, p). We then invert both sets of formulae to obtain the oldcanonical coordinates(q, p) as functions of the newcanonical coordinates(Q, P). Substitution of the inverted formulae into the final equation
The type 3 generating function G3 depends only on the old generalized momenta and the new generalized coordinates
where the terms represent a Legendre transformation to change the left-hand side 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 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
define relations between the new generalized coordinatesQ and the old canonical coordinates(q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates(q, p). We then invert both sets of formulae to obtain the oldcanonical coordinates(q, p) as functions of the newcanonical coordinates(Q, P). Substitution of the inverted formulae into the final equation
The type 4 generating function depends only on the old and new generalized momenta
where the 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 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
define relations between the new generalized momenta P and the old canonical coordinates(q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates(q, p). We then invert both sets of formulae to obtain the oldcanonical coordinates(q, p) as functions of the newcanonical coordinates(Q, P). Substitution of the inverted formulae into the final equation
Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If and , then Hamilton's principle is automatically satisfied
since a valid trajectory should always satisfy Hamilton's principle, regardless of the endpoints.
The translation where are two constant vectors is a canonical transformation. Indeed, the Jacobian matrix is the identity, which is symplectic: .
Set and , the transformation where is a rotation matrix of order 2 is canonical. Keeping in mind that special orthogonal matrices obey it's easy to see that the Jacobian is symplectic. Be aware that this example only works in dimension 2: is the only special orthogonal group in which every matrix is symplectic.
The transformation , where is an arbitrary function of , is canonical. Jacobian matrix is indeed given by
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 coordinatesq is written here as a superscript (), not as a subscript as done above (). 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.
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.