Canonical transformation explained

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 into$$P\_i\; =\; \backslash frac\backslash ,$$where

\left\{ (P _{1} , Q _{1), (P} _{2,} Q _{2), (P} _{3,} Q _{3), \ldots \right\}} are the new co‑ordinates, grouped in canonical conjugate pairs of momenta P _{i} and corresponding positions Q _{i,} for i = 1 , 2 , \ldots N, with N being the number of degrees of freedom in both co‑ordinate 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, e.g.,

A dot over a variable or list signifies the time derivative, e.g.,$$\backslash dot\; \backslash equiv\; \backslash frac$$and the equalities are read to be satisfied for all coordinates, for example:$$\backslash dot\; =\; -\backslash frac\backslash quad\; \backslash Longleftrightarrow\; \backslash quad\; \backslash dot\; =\; -\backslash frac\backslash quad\; (\backslash forall\; i).$$

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.,$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash equiv\; \backslash sum\_^\; p\_\; q\_.$$

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 do not have explicit time dependence, ie. $\backslash mathbf\; Q=\backslash mathbf\; Q(\backslash mathbf\; q,\backslash mathbf\; p)$ and $\backslash mathbf\; P=\backslash mathbf\; P(\backslash mathbf\; q,\backslash mathbf\; p)$. The functional form of Hamilton's equations isIn 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 (sometimes called the Kamiltonian) can be expressed as:$$K(\backslash mathbf\; Q,\; \backslash mathbf\; P,\; t)=\; H(q(\backslash mathbf\; Q,\backslash mathbf\; P),p(\backslash mathbf\; Q,\backslash mathbf\; P),t)\; +\; \backslash frac(t)$$where it differs by a partial time derivative of a function known as 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 dependance.

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, 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:

$$\backslash dot=\; J\; \backslash nabla\_\backslash eta\; H$$

Where

and $\backslash mathbf\; =\; \backslash begin\; q\_1\backslash \backslash \; \backslash vdots\; \backslash \backslash \; q\_n\backslash \backslash \; p\_1\backslash \backslash \; \backslash vdots\backslash \backslash \; p\_n\backslash \backslash \; \backslash end$. Similarly, let $\backslash mathbf\; =\; \backslash begin\; Q\_1\backslash \backslash \; \backslash vdots\; \backslash \backslash \; Q\_n\backslash \backslash \; P\_1\backslash \backslash \; \backslash vdots\backslash \backslash \; P\_n\backslash \backslash \; \backslash end$.

From the relation of partial derivatives, converting the

• η = J \nabla _{η} H relation in terms of partial derivatives with new variables gives • η = J ( M ^{T} \nabla _{\varepsilon} H) where $M\; :=\; \backslash frac$. Similarly for $\backslash dot$,

$$\backslash dot=M\backslash dot\; =M\; J\; M^T\; \backslash nabla\_\backslash varepsilon\; H$$

Due to form of the Hamiltonian equations for $\backslash dot$,

$$\backslash dot=J\; \backslash nabla\_\backslash varepsilon\; K\; =\; J\; \backslash nabla\_\backslash varepsilon\; H$$

where $\backslash nabla\_\backslash varepsilon\; K\; =\; \backslash nabla\_\backslash varepsilon\; H$ can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:

$$M\; J\; M^T\; =\; J$$The left hand side of the above is called the Poisson matrix of

\varepsilon , denoted as $\backslash mathcal\; P(\backslash varepsilon)\; =\; MJM^T$. Similarly, a Lagrange matrix of η can be constructed as $\backslash mathcal\; L(\backslash eta)\; =\; M^TJM$. It can be shown that the symplectic condition is also equivalent to $M^T\; J\; M\; =\; J$ by using the $J^=-J$ property. The set of all matrices $M$ which satisfy symplectic conditions form a symplectic group. The symplectic conditions are equivalent with indirect conditions as they both lead to the equation $\backslash dot=\; J\; \backslash nabla\_\backslash varepsilon\; H$, which is used in both of the derivations.

Invariance of Poisson Bracket

The Poisson bracket which is defined as:$$\backslash \_\backslash eta\; :=\; \backslash sum\_^\; \backslash left(\backslash frac\; \backslash frac\; -\; \backslash frac\; \backslash frac\backslash right)$$can be represented in matrix form as:$$\backslash \_\backslash eta\; :=\; (\backslash nabla\_\backslash eta\; u)^T\; J\; (\backslash nabla\_\backslash eta\; v)$$Hence using partial derivative relations and symplectic condition gives:$$\backslash \_\backslash eta\; =\; (\backslash nabla\_\backslash eta\; u)^T\; J\; (\backslash nabla\_\backslash eta\; v)\; =\; (M^T\; \backslash nabla\_\backslash varepsilon\; u)^T\; J\; (M^T\; \backslash nabla\_\backslash varepsilon\; v)\; =\; (\backslash nabla\_\backslash varepsilon\; u)^T\; M\; J\; M^T\; (\backslash nabla\_\backslash varepsilon\; v)\; =\; (\backslash nabla\_\backslash varepsilon\; u)^T\; J\; (\backslash nabla\_\backslash varepsilon\; v)\; =\; \backslash \_\backslash varepsilon$$

The symplectic condition can also be recovered by taking $u=\backslash varepsilon\_i$ and $v=\backslash varepsilon\_j$ which shows that $(M\; J\; M^T)\_=\; J\_$. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that $\backslash mathcal\; P\_(\backslash varepsilon)\; =\; \backslash \_\backslash eta\; =(M\; J\; M^T)\_$, which is also the result of explicitly calculating the matrix element by expanding it.

Invariance of Lagrange Bracket

The Lagrange bracket which is defined as:

$$[u,\; v\; ]\_\; :=\; \backslash sum\_^n\; \backslash left(\backslash frac\; \backslash frac\; -\; \backslash frac\; \backslash frac\; \backslash right)$$

can be represented in matrix form as:

$$[u,\; v\; ]\_\; :=\; \backslash left(\backslash frac\; \backslash right)^T\; J\; \backslash left(\backslash frac\; \backslash right)$$

Using similar derivation, gives:

$$[u,\; v]\_\backslash varepsilon\; =\; (\backslash partial\_u\; \backslash varepsilon)^T\; \backslash ,J\backslash ,\; (\backslash partial\_v\; \backslash varepsilon)\; =\; (M\; \backslash ,\; \backslash partial\_u\; \backslash eta)^T\; \backslash ,J\; \backslash ,(M\; \backslash ,\backslash partial\_v\; \backslash eta)\; =\; (\backslash partial\_u\; \backslash eta)^T\backslash ,\; M^TJ\; M\backslash ,(\backslash partial\_v\; \backslash eta)\; =\; (\backslash partial\_u\; \backslash eta)^T\backslash ,\; J\backslash ,(\backslash partial\_v\; \backslash eta)\; =\; [u,\; v]\_\backslash eta$$The symplectic condition can also be recovered by taking $u=\backslash eta\_i$ and $v=\backslash eta\_j$ which shows that $(M^T\; J\; M)\_=\; J\_$. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that $\backslash mathcal\; L\_(\backslash eta)\; =[\backslash eta\_i,\backslash eta\_j]\_\backslash varepsilon=(M^T\; J\; M)\_$, 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:

$d\; \backslash varepsilon=(dq\_1,\; dp\_,0,0,\backslash ldots),\backslash quad\backslash delta\; \backslash varepsilon=(\backslash delta\; q\_,\backslash delta\; p\_,0,0,\backslash ldots).$

The area of the infinitesimal parallelogram is given by:

$\backslash delta\; a(12)=d\; q\_\backslash delta\; p\_-\backslash delta\; q\_\; d\; p\_=^T\backslash ,J\; \backslash ,\; d\; \backslash varepsilon.$

It follows from the $M^T\; J\; M\; =\; J$ symplectic condition that the infinitesimal area is conserved under canonical transformation:

$\backslash delta\; a(12)=^T\backslash ,J\; \backslash ,d\; \backslash varepsilon=^T\backslash ,J\; \backslash ,Md\; \backslash eta=\; ^T\backslash ,M^TJM\; \backslash ,d\; \backslash eta\; =\; ^T\backslash ,J\; \backslash ,d\backslash eta\; =\; \backslash delta\; A(12).$

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 $^T\backslash ,J\; \backslash ,\; \backslash delta\; \backslash varepsilon$ under canonical transformation, expanded as:$$\backslash sum\; \backslash delta\; q\; \backslash cdot\; dp\; -\; \backslash delta\; p\; \backslash cdot\; dq\; =\; \backslash sum\; \backslash delta\; Q\; \backslash cdot\; dP\; -\; \backslash delta\; P\; \backslash cdot\; dQ$$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, $^T\backslash ,J\; \backslash ,\; w$ is also known as a symplectic product of the vectors $$ and $w$ 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.,$$\backslash int\; \backslash mathrm\backslash mathbf\backslash ,\; \backslash mathrm\backslash mathbf\; =\; \backslash int\; \backslash mathrm\backslash mathbf\backslash ,\; \backslash mathrm\backslash mathbf$$

By calculus, the latter integral must equal the former times the determinant of Jacobian $$\backslash int\; \backslash mathrm\backslash mathbf\backslash ,\; \backslash mathrm\backslash mathbf\; =\; \backslash int\; \backslash det\; (M)\; \backslash ,\; \backslash mathrm\backslash mathbf\backslash ,\; \backslash mathrm\backslash mathbf$$Where $M\; :=\; \backslash frac$

Exploiting the "division" property of Jacobians yields$$M\; \backslash equiv\; \backslash frac\; \backslash left/\; \backslash frac\; \backslash right.$$

Eliminating the repeated variables gives$$M\; \backslash equiv\; \backslash frac\; \backslash left/\; \backslash frac\; \backslash right.$$

Application of the indirect conditions above yields .

Generating function approach

l{L} _{qp} = p ⋅ • q - H( q , p , t) and l{L} _{QP} = P ⋅ • Q - K( Q , P , t) , 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$$\backslash lambda\; \backslash left[\backslash mathbf\{p\}\; \backslash cdot\; \backslash dot\{\backslash mathbf\{q\}\}\; -\; H(\backslash mathbf\{q\},\; \backslash mathbf\{p\},\; t)\; \backslash right]\; =\; \backslash mathbf\; \backslash cdot\; \backslash dot\; -\; K(\backslash mathbf,\; \backslash mathbf,\; t)\; +\; \backslash frac$$

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:

Generating Function Derivatives!Transformed Hamiltonian! colspan="3"

Trivial Cases
G = G _{1(q,Q,t)}	p = (\partial G _{1})/(\partial q)	P = - (\partial G _{1})/(\partial Q)	$K\; =\; H\; +\; \backslash frac$	G _{1} = qQ	Q = p	P = - q
G = G _{2(q,P,t)} - QP	p = (\partial G _{2})/(\partial q)	Q = (\partial G _{2})/(\partial P)		G _{2} = qP	Q = q	P = p
G = G _{3(p,Q,t)} + qp	q =- (\partial G _{3})/(\partial p)	P =- (\partial G _{3})/(\partial Q)		G _{3} = pQ	Q = - q	P = - p
G = G _{4(p,P,t)} + qp - QP	q =- (\partial G _{4})/(\partial p)	Q = (\partial G _{4})/(\partial P)		G _{4} = pP	Q = p	P = - q

Type 1 generating function

The type 1 generating function depends only on the old and new generalized coordinates$$G\; \backslash equiv\; G\_(\backslash mathbf,\; \backslash mathbf,\; t)$$To derive the implicit transformation, we expand the defining equation above$$\backslash mathbf\; \backslash cdot\; \backslash dot\; -\; H(\backslash mathbf,\; \backslash mathbf,\; t)\; =\; \backslash mathbf\; \backslash cdot\; \backslash dot\; -\; K(\backslash mathbf,\; \backslash mathbf,\; t)\; +\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash dot\; +\; \backslash frac\; \backslash cdot\; \backslash dot$$

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$$\backslash \; \backslash mathbf\; =\; \backslash frac\backslash$$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$$\backslash mathbf\; =\; -\backslash frac$$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 $$K\; =\; H\; +\; \backslash frac$$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 $$G\_\; \backslash equiv\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$$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

The type 2 generating function

G _{2} ( q , P , t) depends only on the old generalized coordinates and the new generalized momenta$$G\; \backslash equiv\; G\_(\backslash mathbf,\; \backslash mathbf,\; t)-\backslash mathbf\; \backslash cdot\; \backslash mathbf$$where the - Q ⋅ P 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$$\backslash mathbf\; \backslash cdot\; \backslash dot\; -\; H(\backslash mathbf,\; \backslash mathbf,\; t)\; =\; -\backslash mathbf\; \backslash cdot\; \backslash dot\; -\; K(\backslash mathbf,\; \backslash mathbf,\; t)\; +\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash dot\; +\; \backslash frac\; \backslash cdot\; \backslash dot$$

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 equations$$\backslash mathbf\; =\; \backslash frac$$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 equations$$\backslash mathbf\; =\; \backslash frac$$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 . Substitution of the inverted formulae into the final equation $$K\; =\; H\; +\; \backslash frac$$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 $$G\_\; \backslash equiv\; \backslash mathbf(\backslash mathbf;\; t)\; \backslash cdot\; \backslash mathbf$$where is a set of functions. This results in a point transformation of the generalized coordinates$$\backslash mathbf\; =\; \backslash frac\; =\; \backslash mathbf(\backslash mathbf;\; t)$$

Type 3 generating function

The type 3 generating function

G _{3} ( p , Q , t) depends only on the old generalized momenta and the new generalized coordinates $$G\; \backslash equiv\; G\_(\backslash mathbf,\; \backslash mathbf,\; t)+\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$$where the q ⋅ p 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$$-\backslash mathbf\; \backslash cdot\; \backslash dot\; -\; H(\backslash mathbf,\; \backslash mathbf,\; t)\; =\; \backslash mathbf\; \backslash cdot\; \backslash dot\; -\; K(\backslash mathbf,\; \backslash mathbf,\; t)\; +\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash dot\; +\; \backslash frac\; \backslash cdot\; \backslash dot$$

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$$\backslash mathbf\; =\; -\backslash frac$$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$$\backslash mathbf\; =\; -\backslash frac$$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 $$K\; =\; H\; +\; \backslash frac$$ 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

The type 4 generating function

G _{4} ( p , P , t) depends only on the old and new generalized momenta$$G\; \backslash equiv\; G\_(\backslash mathbf,\; \backslash mathbf,\; t)\; +\backslash mathbf\; \backslash cdot\; \backslash mathbf\; -\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$$where the q ⋅ p - Q ⋅ P terms represent a Legendre transformation to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above$$-\backslash mathbf\; \backslash cdot\; \backslash dot\; -\; H(\backslash mathbf,\; \backslash mathbf,\; t)\; =\; -\backslash mathbf\; \backslash cdot\; \backslash dot\; -\; K(\backslash mathbf,\; \backslash mathbf,\; t)\; +\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash dot\; +\; \backslash frac\; \backslash cdot\; \backslash dot$$

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$$\backslash mathbf\; =\; -\backslash frac$$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 equations$$\backslash mathbf\; =\; \backslash frac$$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 . Substitution of the inverted formulae into the final equation $$K\; =\; H\; +\; \backslash frac$$yields a formula for as a function of the new canonical coordinates .

Restrictions on generating functions

For example, using generating function of second kind: $\_i\; =\; \backslash frac$ and $\_i\; =\; \backslash frac$, the first set of equations consisting of variables $\backslash mathbf$, $\backslash mathbf$ and $\backslash mathbf$ has to be inverted to get $\backslash mathbf(\backslash mathbf\; q,\; \backslash mathbf\; p)$. This process is possible when the matrix defined by $a\_=\backslash frac$ is non-singular.

Hence, restrictions are placed on generating functions to have the matrices: $\backslash left[\backslash frac\{\backslash partial^2\; G\_1\}\{\backslash partial\; Q\_j\backslash partial\; q\_i\}\; \backslash right]$, $\backslash left[\backslash frac\{\backslash partial^2\; G\_2\}\{\backslash partial\; P\_j\backslash partial\; q\_i\}\; \backslash right]$, $\backslash left[\backslash frac\{\backslash partial^2\; G\_3\}\{\backslash partial\; p\_j\backslash partial\; Q\_i\}\; \backslash right]$ and $\backslash left[\backslash frac\{\backslash partial^2\; G\_4\}\{\backslash partial\; p\_j\backslash partial\; P\_i\}\; \backslash right]$, being non-singular.

Limitations of generating functions

Since $\backslash left[\backslash frac\{\backslash partial^2\; G\_2\}\{\backslash partial\; P\_j\backslash partial\; q\_i\}\; \backslash right]$ is non-singular, it implies that $\backslash left[\backslash frac\{\backslash partial\; \{p\}\_i(\backslash mathbf\; q,\backslash mathbf\; P)\}\{\backslash partial\; P\_j\}\backslash right]$ is also non-singular. Since the matrix $\backslash left[\backslash frac\{\backslash partial\; P\_i(\backslash mathbf\; q,\backslash mathbf\; p)\}\{\backslash partial\; p\_j\}\; \backslash right]$ is inverse of $\backslash left[\backslash frac\{\backslash partial\; \{p\}\_i(\backslash mathbf\; q,\backslash mathbf\; P)\}\{\backslash partial\; P\_j\}\backslash right]$, the transformations of type 2 generating functions always have a non-singular $\backslash left[\backslash frac\{\backslash partial\; P\_i(\backslash mathbf\; q,\backslash mathbf\; p)\}\{\backslash partial\; p\_j\}\; \backslash right]$ matrix. Similarly, it can be stated that type 1 and type 4 generating functions always have a non-singular $\backslash left[\backslash frac\{\backslash partial\; Q\_i(\backslash mathbf\; q,\backslash mathbf\; p)\}\{\backslash partial\; p\_j\}\; \backslash right]$ matrix whereas type 2 and type 3 generating functions always have a non-singular $\backslash left[\backslash frac\{\backslash partial\; P\_i(\backslash mathbf\; q,\backslash mathbf\; p)\}\{\backslash partial\; p\_j\}\; \backslash right]$ matrix. Hence, the canonical transformations resulting from these generating functions are not completely general.

In other words, since and are each independent functions, it follows that to have generating function of the form $G\_(\backslash mathbf,\; \backslash mathbf,\; t)$ and

G _{4} ( p , P , t) or G _{2} ( q , P , t) and G _{3} ( p , Q , t) , the corresponding Jacobian matrices $\backslash left[\backslash frac\{\backslash partial\; Q\_i\}\{\backslash partial\; p\_j\}\; \backslash right]$ and $\backslash left[\backslash frac\{\backslash partial\; P\_i\}\{\backslash partial\; p\_j\}\; \backslash right]$ 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 proved 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

From:

K = H + (\partial G)/(\partial t) , calculate $\backslash frac$:

$$\backslash begin\backslash left(\backslash frac\backslash right)\_=\; \backslash frac\; -\; \backslash frac\backslash frac\; -\; \backslash frac\backslash frac\; -\; \backslash frac\backslash left(\backslash frac\backslash right)\_\; =\; \backslash dot\; +\; \backslash dot\; \backslash frac\; -\; \backslash dot\backslash frac\; \backslash \backslash =\; \backslash frac\; +\; \backslash frac\; \backslash cdot\; \backslash dot\; +\; \backslash frac\; \backslash cdot\; \backslash dot\; +\; \backslash dot\; \backslash frac\; -\; \backslash dot\backslash frac\; \backslash \backslash =\backslash dot\backslash left(\backslash frac\; -\; \backslash frac\backslash right)+\backslash dot\backslash left(\backslash frac\; +\backslash frac\; \backslash right)\; +\; \backslash frac\; \backslash end$$Since the left hand side is $\backslash frac\; =\; \backslash frac\; \backslash partial\; \backslash left(\backslash frac\; \backslash right)\; \backslash bigg\; |\_$ which is independent of dynamics of the particles, equating coefficients of $\backslash dot\; q$ and $\backslash dot\; p$ to zero, canonical transformation rules are obtained. This step is equivalent to equating the left hand side as $\backslash frac\; =\; \backslash frac$.

Similarly:

$$\backslash begin\backslash left(\backslash frac\backslash right)\_=\; \backslash frac\; -\; \backslash frac\backslash frac\; -\; \backslash frac\backslash frac\; -\; \backslash frac\backslash left(\backslash frac\backslash right)\_\; =\; -\backslash dot\; +\; \backslash dot\; \backslash frac\; -\; \backslash dot\backslash frac\; \backslash \backslash =\; -\backslash frac\; -\backslash frac\; \backslash cdot\; \backslash dot\; -\; \backslash frac\; \backslash cdot\; \backslash dot\; +\; \backslash dot\; \backslash frac\; -\; \backslash dot\backslash frac\; \backslash \backslash =-\backslash left(\backslash dot\backslash left(\backslash frac\; +\; \backslash frac\backslash right)+\backslash dot\backslash left(\backslash frac\; -\backslash frac\; \backslash right)\; +\; \backslash frac\; \backslash right)\backslash end$$Similarly the canonical transformation rules are obtained by equating the left hand side as $\backslash frac\; =\; -\; \backslash frac$.

The above two relations can be combined in matrix form as: $J\; \backslash left(\backslash nabla\_\backslash varepsilon\; \backslash frac\; \backslash right)\; =\; \backslash frac$ (which will also retain same form for extended canonical transformation) where the result $\backslash frac\; =\; K-H$, has been used. The canonical transformation relations are hence said to be equivalent to $J\; \backslash left(\backslash nabla\_\backslash varepsilon\; \backslash frac\; \backslash right)\; =\; \backslash frac$ in this context.

The canonical transformation relations can now be restated to include time dependance:Since $\backslash frac\; =\; \backslash frac$ and $\backslash frac\; =\; -\; \backslash frac$, if and do not explicitly depend on time, $K=\; H\; +\; \backslash frac(t)$ can be taken. The analysis of restricted canonical transformations is hence consistent with this generalization.

Symplectic Condition

Applying transformation of co-ordinates formula for

\nabla _{η} H = M ^{T} \nabla _{\varepsilon} H , in Hamiltonian's equations gives:$$\backslash dot=J\backslash nabla\_\backslash eta\; H\; =J\; (M^T\; \backslash nabla\_\backslash varepsilon\; H)$$

Similarly for $\backslash dot$:$$\backslash dot=M\backslash dot\; +\; \backslash frac\; =M\; J\; M^T\; \backslash nabla\_\backslash varepsilon\; H\; +\; \backslash frac$$or:$$\backslash dot=J\; \backslash nabla\_\backslash varepsilon\; K\; =\; J\; \backslash nabla\_\backslash varepsilon\; H\; +\; J\; \backslash nabla\_\backslash varepsilon\; \backslash left(\backslash frac\backslash right)$$Where the last terms of each equation cancel due to $J\; \backslash left(\backslash nabla\_\backslash varepsilon\; \backslash frac\; \backslash right)\; =\; \backslash frac$ condition from canonical transformations. Hence leaving the symplectic relation: $M\; J\; M^T\; =\; J$ which is also equivalent with the condition $M^T\; J\; M\; =\; J$. It follows from the above two equations that the symplectic condition implies the equation $J\; \backslash left(\backslash nabla\_\backslash varepsilon\; \backslash frac\; \backslash right)\; =\; \backslash frac$, 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 Poisson and Lagrange Bracket

Since $\backslash mathcal\; P\_(\backslash varepsilon)\; =\; \backslash \_\backslash eta\; =(M\; J\; M^T)\_\; =\; J\_$ and $\backslash mathcal\; L\_(\backslash eta)\; =[\backslash eta\_i,\backslash eta\_j]\_\backslash varepsilon=(M^T\; J\; M)\_\; =\; J\_$ where the symplectic condition is used in the last equalities. Using $\backslash \_\backslash varepsilon=[\backslash eta\_i,\backslash eta\_j]\_\backslash eta\; =\; J\_$, the equalities $\backslash \_\backslash eta=\; \backslash \_\backslash varepsilon$ and $[\backslash eta\_i,\backslash eta\_j]\_\backslash varepsilon=\; [\backslash eta\_i,\backslash eta\_j]\_\backslash eta$ are obtained which imply the invariance of Poisson and Lagrange brackets.

Extended Canonical Transformation

Canonical transformation relations

By solving for:$$\backslash lambda\; \backslash left[\backslash mathbf\{p\}\; \backslash cdot\; \backslash dot\{\backslash mathbf\{q\}\}\; -\; H(\backslash mathbf\{q\},\; \backslash mathbf\{p\},\; t)\; \backslash right]\; =\; \backslash mathbf\; \backslash cdot\; \backslash dot\; -\; K(\backslash mathbf,\; \backslash mathbf,\; t)\; +\; \backslash frac$$with various forms of generating function, the relation between K and H goes as $\backslash frac\; =\; K-\backslash lambda\; H$ instead, which also applies for $\backslash lambda\; =\; 1$ case.

All results presented below can also be obtained by replacing $q\; \backslash rightarrow\; \backslash sqrtq$, $p\; \backslash rightarrow\; \backslash sqrtp$ and $H\; \backslash rightarrow\; H$ from known solutions, since it retains the form of Hamilton's equations. The extended canonical transformations are hence said to be result of a canonical transformation ($\backslash lambda\; =\; 1$) and a trivial canonical transformation ($\backslash lambda\; \backslash neq\; 1$) which has $M\; J\; M^T\; =\; \backslash lambda\; J$ (for the given example, $M\; =\; \backslash sqrt\; I$ which satisfies the condition).

Using same steps previously used in previous generalization, with $\backslash frac\; =\; K-\backslash lambda\; H$ in the general case, and retaining the equation $J\; \backslash left(\backslash nabla\_\backslash varepsilon\; \backslash frac\; \backslash right)\; =\; \backslash frac$, extended canonical transformation partial differential relations are obtained as:

Symplectic condition

Following the same steps to derive the symplectic conditions, as: $$\backslash dot=J\backslash nabla\_\backslash eta\; H\; =J\; (M^T\; \backslash nabla\_\backslash varepsilon\; H)$$and $$\backslash dot=M\backslash dot\; +\; \backslash frac\; =M\; J\; M^T\; \backslash nabla\_\backslash varepsilon\; H\; +\; \backslash frac$$

where using $\backslash frac\; =\; K-\backslash lambda\; H$ instead gives:$$\backslash dot=J\; \backslash nabla\_\backslash varepsilon\; K\; =\; \backslash lambda\; J\; \backslash nabla\_\backslash varepsilon\; H\; +\; J\; \backslash nabla\_\backslash varepsilon\; \backslash left(\backslash frac\backslash right)$$The second part of each equation cancel. Hence the condition for extended canonical transformation instead becomes: $M\; J\; M^T\; =\; \backslash lambda\; J$.

Poisson and Lagrange Brackets

The Poisson brackets are changed as follows:$$\backslash \_\backslash eta\; =\; (\backslash nabla\_\backslash eta\; u)^T\; J\; (\backslash nabla\_\backslash eta\; v)\; =\; (M^T\; \backslash nabla\_\backslash varepsilon\; u)^T\; J\; (M^T\; \backslash nabla\_\backslash varepsilon\; v)\; =\; (\backslash nabla\_\backslash varepsilon\; u)^T\; M\; J\; M^T\; (\backslash nabla\_\backslash varepsilon\; v)\; =\; \backslash lambda\; (\backslash nabla\_\backslash varepsilon\; u)^T\; J\; (\backslash nabla\_\backslash varepsilon\; v)\; =\; \backslash lambda\; \backslash \_\backslash varepsilon$$whereas, the Lagrange brackets are changed as:

$$[u,\; v]\_\backslash varepsilon\; =\; (\backslash partial\_u\; \backslash varepsilon)^T\; \backslash ,J\backslash ,\; (\backslash partial\_v\; \backslash varepsilon)\; =\; (M\; \backslash ,\; \backslash partial\_u\; \backslash eta)^T\; \backslash ,J\; \backslash ,(M\; \backslash ,\backslash partial\_v\; \backslash eta)\; =\; (\backslash partial\_u\; \backslash eta)^T\backslash ,\; M^TJ\; M\backslash ,(\backslash partial\_v\; \backslash eta)\; =\; \backslash lambda\; (\backslash partial\_u\; \backslash eta)^T\backslash ,\; J\backslash ,(\backslash partial\_v\; \backslash eta)\; =\; \backslash lambda\; [u,\; v]\_\backslash eta$$Hence, the Poisson bracket scales by the inverse of $\backslash lambda$ whereas the Lagrange bracket scales by a factor of $\backslash lambda$.^[1]

Infinitesimal canonical transformation

Consider the canonical transformation that depends on a continuous parameter

\alpha , as follows:\begin{align} & Q(q,p,t;\alpha) & Q(q,p,t; 0 ) = q \ & P(q,p,t;\alpha) with & P(q,p,t; 0 ) = p \ \end{align}

For infinitesimal values of

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

Consider the following generating function:

G _{2(q,P,t)=} qP + \alpha G(q,P,t)

Since for

\alpha =0 , G _{2} = qP has the resulting canonical transformation, Q = q and P = p , this type of generating function can be used for infinitesimal canonical transformation by restricting \alpha to an infinitesimal value. From the conditions of generators of second type:Since P = P(q,p,t;\alpha) , changing the variables of the function G to G(q,p,t) and neglecting terms of higher order of \alpha , gives:Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition.^[2]

Active canonical transformations

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

\begin{align} & \delta q = \alpha (\partial G)/(\partial p) (q,p,t) and \delta p = - \alpha (\partial G)/(\partial q) (q,p,t) , \ \end{align}

or as

\delta η = \alpha J \nabla _{η} G in matrix form.

For any function

u( η ) , it changes under active view of the transformation according to:\delta u = u( η + \delta η ) - u( η ) = (\nabla _{η} u) ^{T\deltaη=\alpha} (\nabla _{η} u) ^{T} J (\nabla _{η} G) = \alpha \{ u,G \} .

Considering the change of Hamiltonians in the active view, ie. for a fixed point,$$K(Q=q\_0,P=p\_0,t)\; -\; H(q=q\_0,p=p\_0,t)\; =\; \backslash left(H(q\_0\text{'},p\_0\text{'},t)\; +\; \backslash frac\backslash right)\; -\; H(q\_0,p\_0,t)\; =\; -\; \backslash delta\; H\; +\backslash alpha\; \backslash frac\; =\; \backslash alpha\backslash left(\backslash +\backslash frac\; \backslash right)=\backslash alpha\backslash frac$$where $(q=q\_0\text{'},p=p\_0\text{'})$ are mapped to the point, $(Q=q\_0,P=p\_0)$ by the infinitesimal canonical transformation, and similar change of variables for

G(q,P,t) to G(q,p,t) is considered up-to first order of \alpha . Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.

Examples of ICT

Time evolution

Taking

G(q,p,t) = H(q,p,t) and \alpha = dt , then \delta η = (J \nabla _{η} H) dt = • η dt = d η . Thus the continuous application of such a transformation maps the coordinates η (\tau) to η (\tau + t) . Hence if the Hamiltonian is time translation invariant, i.e. does not have explicit time dependance, its value is conserved for the motion.

Translation

Taking

G(q,p,t) = p _{k} , \delta p _{i} = 0 and \delta q _{i} = \alpha \delta _{ik} . 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:

\begin{array}{l}{{ q=\left(x _{1} ,y _{1} ,z _{1} ,\ldots,x _{n} ,y _{n} ,z _{n} \right),}}\ {{ p=\left(p _{1x} ,p _{1y} ,p _{1z} ,\ldots,p _{n} ,p _{n} ,p _{n} \right).}}\end{array}

Choosing the generator to be:

G = L _{z} = \sum ^{n} _{i =1} \left(x _{i} p _{i} - y _{i} p _{i} \right) and the infinitesimal value of \alpha = \delta \phi , then the change in the coordinates is given for x by:\begin{array}{c} {\delta x _{i} = \{x _{i} ,G\}\delta\phi = \displaystyle\sum _{j} \{x _{i} ,x _{j} p _{j} - y _{j} p _{j} \}\delta\phi = \displaystyle\sum _{j} (\underbrace{\{x _{i} ,x _{j} p _{j} \}} _{=0} - { \{x _{i} ,y _{j} p _{j} \}}})\delta\phi\\ {{ = \displaystyle - \sum _{j} y _{j} \underbrace{\{x _{i,p} _{jx} \}} _{= \delta _{ij}} \delta\phi =- y _{i} \delta \phi}} \end{array}

and similarly for y:

\begin{array}{c} \delta y _{i} = \{y _{i} ,G\}\delta\phi = \displaystyle\sum _{j} \{y _{i} ,x _{j} p _{j} - y _{j} p _{j} \}\delta\phi = \displaystyle\sum _{j} (\{y _{i} ,x _{j} p _{j} \} - \underbrace{ \{y _{i} ,y _{j} p _{j} \}} _{=0} )\delta \phi\ { = \displaystyle\sum _{j} x _{j} \underbrace{\{y _{i,p} _{jy} \}} _{= \delta _{ij}} \delta\phi = x _{i} \delta\phi,} \end{array}

whereas the z component of all particles is unchanged:

\delta z = \left\{z _{i} ,G\right\}\delta\phi = \sum _{j} \left\{z _{i} ,x _{j} p _{j} - y _{j} p _{j} \right\}\delta \phi =0 .

These transformations correspond to rotation about the z axis by angle

\delta \phi 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.

Motion as canonical transformation

Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If

Q (t) \equiv q (t + \tau) and P (t) \equiv p (t + \tau) , then Hamilton's principle is automatically satisfied$$\backslash delta\; \backslash int\_^\; \backslash left[\backslash mathbf\{P\}\; \backslash cdot\; \backslash dot\{\backslash mathbf\{Q\}\}\; -\; K(\backslash mathbf\{Q\},\; \backslash mathbf\{P\},\; t)\; \backslash right]\; dt\; =\; \backslash delta\; \backslash int\_^\; \backslash left[\backslash mathbf\{p\}\; \backslash cdot\; \backslash dot\{\backslash mathbf\{q\}\}\; -\; H(\backslash mathbf\{q\},\; \backslash mathbf\{p\},\; t+\backslash tau)\; \backslash right]\; dt\; =\; 0$$since a valid trajectory ( q (t), p (t)) should always satisfy Hamilton's principle, regardless of the endpoints.

Examples

• The translation

Q ( q , p ) = q + a , P ( q , p ) = p + b where a , b are two constant vectors is a canonical transformation. Indeed, the Jacobian matrix is the identity, which is symplectic: I ^{TJI=J} .

• Set

x = (q,p) and X = (Q,P) , the transformation X ( x ) = R x where R \in SO( 2 ) is a rotation matrix of order 2 is canonical. Keeping in mind that special orthogonal matrices obey R ^{TR=I} it's easy to see that the Jacobian is symplectic. However, this example only works in dimension 2: SO( 2 ) is the only special orthogonal group in which every matrix is symplectic. Note that the rotation here acts on (q,p) and not on q and p independently, so these are not the same as a physical rotation of an orthogonal spatial coordinate system.

• The transformation

(Q(q,p), P(q,p)) = (q + f(p), p) , where f(p) is an arbitrary function of p , is canonical. Jacobian matrix is indeed given by 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 to be written as$$\backslash sum\_i\; p\_i\backslash ,dq^i$$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 (

q ^{i} ), not as a subscript as done above (q _{i} ). 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

References

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• Book: Giacaglia, Georgio Eugenio Oscare . Perturbation Methods in Non-Linear Systems . Springer-Verlag . 1972 . 3-540-90054-3 . New York . 72-87714.
• Book: Lanczos, Cornelius . The Variational Principles of Mechanics . 2012-04-24 . Courier Corporation . 978-0-486-13470-3 . en.
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• Book: Hand . Louis N. . Analytical Mechanics . Finch . Janet D. . Cambridge University Press . 1999 . 978-0521573276 . 1st .
• Book: Sudarshan . E C George . Classical Dynamics: A Modern Perspective . Mukunda . N . Wiley . 2010 . 9780471835400.

Notes and References

1. -17
2. Web site: Hergert . Heiko . December 10, 2021 . PHY422/820: Classical Mechanics . live . https://web.archive.org/web/20231222161338/https://people.nscl.msu.edu/~hergert/phy820/material/pdfs/w14.pdf . December 22, 2023 . December 22, 2023.