Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the 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 q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if we simultaneously change the momentum by a Legendre transformation into $$P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}.$$

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.

Notation

Boldface variables such as q represent a list of N generalized coordinates that need not transform like a vector under rotation, e.g., $$\mathbf {q} \equiv \left(q_{1},q_{2},\ldots ,q_{N-1},q_{N}\right).$$

A dot over a variable or list signifies the time derivative, e.g., $${\dot {\mathbf {q} }}\equiv {\frac {d\mathbf {q} }{dt}}.$$

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., $$\mathbf {p} \cdot \mathbf {q} \equiv \sum _{k=1}^{N}p_{k}q_{k}.$$

The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value.

Conditions for Restricted Canonical Transformation

This section initially only concerns transformation where Q and P do not have explicit time dependance, ie. $Q=Q(q,p)$ and $P=P(q,p)$. However, concepts developed here can be generalized to include those transformations excluding the bilinear invariance condition. The following conditions are equivalent conditions with the exception of bilinear invariance condition which is only equivalent (as well as applicable) under restricted canonical transformations.

Indirect conditions

The functional form of Hamilton's equations is $${\begin{aligned}{\dot {\mathbf {p} }}&=-{\frac {\partial H}{\partial \mathbf {q} }}\\{\dot {\mathbf {q} }}&={\frac {\partial H}{\partial \mathbf {p} }}\end{aligned}}$$Upon transformation to new coordinates $Q=Q(q,p)$ and $P=P(q,p)$, a new Hamiltonian (sometimes called the Kamiltonian) has to be considered instead:

$$K(\mathbf {Q} ,\mathbf {P} ,t)=H(q(\mathbf {Q} ,\mathbf {P} ),p(\mathbf {Q} ,\mathbf {P} ),t)+{\frac {\partial G}{\partial t}}(t).$$ The reason for choosing the above form of Kamiltonian in this case, will be justified in the generalization. Essentially the new Hamiltonian can differ from the old Hamiltonian by a function purely of time but this only applies for restricted canonical transformations. This allows us to employ the following relations in the derivation:$${\begin{aligned}\left({\frac {\partial H}{\partial q}}\right)_{\mathbf {q} ,\mathbf {p} ,t}=\left({\frac {\partial K}{\partial q}}\right)_{\mathbf {q} ,\mathbf {p} ,t}\\\left({\frac {\partial H}{\partial p}}\right)_{\mathbf {q} ,\mathbf {p} ,t}=\left({\frac {\partial K}{\partial p}}\right)_{\mathbf {q} ,\mathbf {p} ,t}\end{aligned}}$$These equations, combined with the form of Hamilton's equations are sufficient to derive the indirect conditions.

By definition, the transformed coordinates have analogous dynamics

$${\begin{aligned}{\dot {\mathbf {P} }}&=-{\frac {\partial K}{\partial \mathbf {Q} }}\\{\dot {\mathbf {Q} }}&={\frac {\partial K}{\partial \mathbf {P} }}\end{aligned}}$$

where K(Q, P) is the new Hamiltonian that is considered.

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 Q_m is $${\begin{aligned}{\dot {Q}}_{m}&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}\\&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\frac {\partial H}{\partial \mathbf {p} }}-{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\frac {\partial H}{\partial \mathbf {q} }}\\&=\lbrace Q_{m},H\rbrace \end{aligned}}$$ where {⋅, ⋅} is the Poisson bracket.

We also have the identity for the conjugate momentum P_m $${\begin{aligned}&{\frac {\partial K(\mathbf {Q} ,\mathbf {P} ,t)}{\partial P_{m}}}\\&={\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\\&={\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\\&={\frac {\partial H}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\end{aligned}}$$

If the transformation is canonical, these two must be equal, resulting in the equations $${\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}$$

The analogous argument for the generalized momenta P_m leads to two other sets of equations $${\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}$$

These are the indirect conditions to check whether a given transformation is canonical.

Symplectic condition

Sometimes the Hamiltonian relations are represented as:

$${\dot {\eta }}=J\nabla _{\eta }H$$

Where ${\textstyle J:={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},}$

and ${\textstyle \mathbf {\eta } ={\begin{bmatrix}q_{1}\\\vdots \\q_{n}\\p_{1}\\\vdots \\p_{n}\\\end{bmatrix}}}$. Similarly, let ${\textstyle \mathbf {\varepsilon } ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{n}\\P_{1}\\\vdots \\P_{n}\\\end{bmatrix}}}$.

From the relation of partial derivatives, we convert ${\dot {\eta }}=J\nabla _{\eta }H$ relation in terms of partial derivatives with new variables:

${\dot {\eta }}=J(M^{T}\nabla _{\varepsilon }H)$ where ${\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}$.

Similarly we find:$${\dot {\varepsilon }}=M{\dot {\eta }}=MJM^{T}\nabla _{\varepsilon }H$$or since ${\textstyle \nabla _{\varepsilon }K=\nabla _{\varepsilon }H}$ due to the form of Kamiltonian:$${\dot {\varepsilon }}=J\nabla _{\varepsilon }K=J\nabla _{\varepsilon }H$$

we get the symplectic condition:

$$MJM^{T}=J$$It can be shown that this condition is equivalent to satisfying indirect conditions.

Invariance of Poisson Bracket

The Poisson bracket which is defined as:$$\{u,v\}_{\eta }:=\sum _{i=1}^{n}\left({\frac {\partial u}{\partial q_{i}}}{\frac {\partial v}{\partial p_{i}}}-{\frac {\partial u}{\partial p_{i}}}{\frac {\partial v}{\partial q_{i}}}\right)$$can be represented in matrix form as:$$\{u,v\}_{\eta }:=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)$$Hence using partial derivative relations and symplectic condition, we get:$$\{u,v\}_{\eta }=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)=(M^{T}\nabla _{\varepsilon }u)^{T}J(M^{T}\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}MJM^{T}(\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}J(\nabla _{\varepsilon }v)=\{u,v\}_{\varepsilon }$$

Since the equality is expected to hold for any functions, by choice of u and v, we can either recover indirect condition or recover the symplectic condition by showing ${\textstyle (MJM^{T})_{ij}=J_{ij}}$. Thus these conditions are equivalent to symplectic conditions.

Invariance of Lagrange Bracket

The Lagrange bracket is defined as:

$$[u,v]_{\eta }=\sum _{i=1}^{n}\left({\frac {\partial q_{i}}{\partial u}}{\frac {\partial p_{i}}{\partial v}}-{\frac {\partial p_{i}}{\partial u}}{\frac {\partial q_{i}}{\partial v}}\right)$$

Hence the calculated values: ${\textstyle [q_{i},p_{j}]_{\eta }=\delta _{i,j}}$, ${\textstyle [q_{i},q_{j}]_{\eta }=0}$ and ${\textstyle [p_{i},p_{j}]_{\eta }=0}$.

If a matrix were defined as ${\textstyle (A)_{ij}=[\eta _{i},\eta _{j}]_{\eta }}$, then from the above relation, ${\textstyle A={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}}=J}$

The matrix elements of ${\textstyle MJM^{T}}$ can be explicitly calculated to be: ${\textstyle (MJM^{T})_{ij}=[\eta _{i},\eta _{j}]_{\varepsilon }}$

Since ${\textstyle MJM^{T}=J=A}$, it implies ${\textstyle [\eta _{i},\eta _{j}]_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }}$ and hence for arbitrary functions we have: ${\textstyle [u,v]_{\varepsilon }=[u,v]_{\eta }}$. Since the symplectic condition can be trivially recovered from this, the condition serves as an equivalent condition for canonical transformation.

Bilinear invariance conditions for restricted canonical transformation

These set of conditions only apply to restricted canonical transformations or canonical transformations that are independent of time variable.

Due to lack of time dependance in the transformation:

${\textstyle \delta \mathbf {p} ={\frac {\partial \mathbf {p} }{\partial \mathbf {Q} }}\cdot \delta \mathbf {Q} +{\frac {\partial \mathbf {p} }{\partial \mathbf {P} }}\cdot \delta \mathbf {P} }$

${\textstyle \delta \mathbf {q} ={\frac {\partial \mathbf {q} }{\partial \mathbf {Q} }}\cdot \delta \mathbf {Q} +{\frac {\partial \mathbf {q} }{\partial \mathbf {P} }}\cdot \delta \mathbf {P} }$

${\textstyle \delta \mathbf {Q} ={\frac {\partial \mathbf {Q} }{\partial \mathbf {q} }}\cdot \delta \mathbf {q} +{\frac {\partial \mathbf {Q} }{\partial \mathbf {p} }}\cdot \delta \mathbf {p} }$

${\textstyle \delta \mathbf {P} ={\frac {\partial \mathbf {P} }{\partial \mathbf {q} }}\cdot \delta \mathbf {q} +{\frac {\partial \mathbf {P} }{\partial \mathbf {p} }}\cdot \delta \mathbf {p} }$

where similar equations follow for ${\textstyle dq}$, ${\textstyle dp}$, ${\textstyle dQ}$ and ${\textstyle dP}$.

Substituting partial derivatives from canonical transformation conditions, we can show using canonical transformation partial derivative relations that:$$\sum \delta q\cdot dp-\delta p\cdot dq=\sum \delta Q\cdot dP-\delta P\cdot dQ$$If the above is obeyed for any arbitrary variation, it would be only possible if the indirect conditions are met.

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., $$\int \mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} =\int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P}$$

By calculus, the latter integral must equal the former times the determinant of Jacobian M$$\int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P} =\int \det(M)\,\mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p}$$Where ${\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}$

Exploiting the "division" property of Jacobians yields$$M\equiv {\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {P} )}}\left/{\frac {\partial (\mathbf {q} ,\mathbf {p} )}{\partial (\mathbf {q} ,\mathbf {P} )}}\right.$$

Eliminating the repeated variables gives$$M\equiv {\frac {\partial (\mathbf {Q} )}{\partial (\mathbf {q} )}}\left/{\frac {\partial (\mathbf {p} )}{\partial (\mathbf {P} )}}\right.$$

Application of the indirect conditions above yields $\operatorname {det} (M)=1$.

Generating function approach

Main article: Generating function (physics)

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 ${\mathcal {L}}_{qp}=\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)$ and ${\mathcal {L}}_{QP}=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)$ 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): $${\begin{aligned}\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]dt&=0\\\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)\right]dt&=0\end{aligned}}$$

One way for both variational integral equalities to be satisfied is to have $$\lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt}}$$

Lagrangians are not unique: one can always multiply by a constant λ and add a total time derivative ⁠dG/dt⁠ and yield the same equations of motion (as discussed on Wikibooks). 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 various generating functions and its properties tabulated below is discussed in detail:

Properties of four basic Canonical Transformations

Generating Function	Generating Function Derivatives		Trivial Cases
$G=G_{1}(q,Q,t)$	$p={\frac {\partial G_{1}}{\partial q}}$	$P=-{\frac {\partial G_{1}}{\partial Q}}$	$G_{1}=qQ$	$Q=p$	$P=-q$
$G=G_{2}(q,P,t)-QP$	$p={\frac {\partial G_{2}}{\partial q}}$	$Q={\frac {\partial G_{2}}{\partial P}}$	$G_{2}=qP$	$Q=q$	$P=p$
$G=G_{3}(p,Q,t)-qp$	$q=-{\frac {\partial G_{3}}{\partial p}}$	$P=-{\frac {\partial G_{3}}{\partial Q}}$	$G_{3}=pQ$	$Q=-q$	$P=-p$
$G=G_{4}(p,P,t)+qp-QP$	$q=-{\frac {\partial G_{4}}{\partial p}}$	$Q={\frac {\partial G_{4}}{\partial P}}$	$G_{1}=pP$	$Q=p$	$P=-q$

Type 1 generating function

The type 1 generating function G₁ depends only on the old and new generalized coordinates $$G\equiv G_{1}(\mathbf {q} ,\mathbf {Q} ,t)$$ To derive the implicit transformation, we expand the defining equation above $$\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{1}}{\partial t}}+{\frac {\partial G_{1}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial G_{1}}{\partial \mathbf {Q} }}\cdot {\dot {\mathbf {Q} }}$$

Since the new and old coordinates are each independent, the following 2N + 1 equations must hold $${\begin{aligned}\mathbf {p} &={\frac {\partial G_{1}}{\partial \mathbf {q} }}\\\mathbf {P} &=-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}\\K&=H+{\frac {\partial G_{1}}{\partial t}}\end{aligned}}$$

These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations $$\mathbf {p} ={\frac {\partial G_{1}}{\partial \mathbf {q} }}$$ define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Q_k as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations $$\mathbf {P} =-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}$$ 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 old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation $$K=H+{\frac {\partial G_{1}}{\partial t}}$$ yields a formula for K as a function of the new canonical coordinates (Q, P).

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let $$G_{1}\equiv \mathbf {q} \cdot \mathbf {Q}$$ This results in swapping the generalized coordinates for the momenta and vice versa $${\begin{aligned}\mathbf {p} &={\frac {\partial G_{1}}{\partial \mathbf {q} }}=\mathbf {Q} \\\mathbf {P} &=-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}=-\mathbf {q} \end{aligned}}$$ and K = H. 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 G₂ depends only on the old generalized coordinates and the new generalized momenta $$G\equiv G_{2}(\mathbf {q} ,\mathbf {P} ,t)-\mathbf {Q} \cdot \mathbf {P}$$ where the $-\mathbf {Q} \cdot \mathbf {P}$ 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 $$\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)=-\mathbf {Q} \cdot {\dot {\mathbf {P} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{2}}{\partial t}}+{\frac {\partial G_{2}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial G_{2}}{\partial \mathbf {P} }}\cdot {\dot {\mathbf {P} }}$$

Since the old coordinates and new momenta are each independent, the following 2N + 1 equations must hold $${\begin{aligned}\mathbf {p} &={\frac {\partial G_{2}}{\partial \mathbf {q} }}\\\mathbf {Q} &={\frac {\partial G_{2}}{\partial \mathbf {P} }}\\K&=H+{\frac {\partial G_{2}}{\partial t}}\end{aligned}}$$

These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations $$\mathbf {p} ={\frac {\partial G_{2}}{\partial \mathbf {q} }}$$ 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 P_k as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations $$\mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}$$ 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 old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation $$K=H+{\frac {\partial G_{2}}{\partial t}}$$ yields a formula for K as a function of the new canonical coordinates (Q, P).

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let $$G_{2}\equiv \mathbf {g} (\mathbf {q} ;t)\cdot \mathbf {P}$$ where g is a set of N functions. This results in a point transformation of the generalized coordinates $$\mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}=\mathbf {g} (\mathbf {q} ;t)$$

Type 3 generating function

The type 3 generating function G₃ depends only on the old generalized momenta and the new generalized coordinates $$G\equiv G_{3}(\mathbf {p} ,\mathbf {Q} ,t)+\mathbf {q} \cdot \mathbf {p}$$ where the $\mathbf {q} \cdot \mathbf {p}$ 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 $$-\mathbf {q} \cdot {\dot {\mathbf {p} }}-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{3}}{\partial t}}+{\frac {\partial G_{3}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}+{\frac {\partial G_{3}}{\partial \mathbf {Q} }}\cdot {\dot {\mathbf {Q} }}$$

Since the new and old coordinates are each independent, the following 2N + 1 equations must hold $${\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{3}}{\partial \mathbf {p} }}\\\mathbf {P} &=-{\frac {\partial G_{3}}{\partial \mathbf {Q} }}\\K&=H+{\frac {\partial G_{3}}{\partial t}}\end{aligned}}$$

These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations $$\mathbf {q} =-{\frac {\partial G_{3}}{\partial \mathbf {p} }}$$ define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Q_k as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations $$\mathbf {P} =-{\frac {\partial G_{3}}{\partial \mathbf {Q} }}$$ 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 old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation $$K=H+{\frac {\partial G_{3}}{\partial t}}$$ yields a formula for K as a function of the new canonical coordinates (Q, P).

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

Since the new and old coordinates are each independent, the following 2N + 1 equations must hold $${\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{4}}{\partial \mathbf {p} }}\\\mathbf {Q} &={\frac {\partial G_{4}}{\partial \mathbf {P} }}\\K&=H+{\frac {\partial G_{4}}{\partial t}}\end{aligned}}$$

These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations $$\mathbf {q} =-{\frac {\partial G_{4}}{\partial \mathbf {p} }}$$ 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 P_k as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations $$\mathbf {Q} ={\frac {\partial G_{4}}{\partial \mathbf {P} }}$$ 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 old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation $$K=H+{\frac {\partial G_{4}}{\partial t}}$$ yields a formula for K as a function of the new canonical coordinates (Q, P).

Generalization of Canonical transformation conditions

Canonical transformation relations

From: $K=H+{\frac {\partial G}{\partial t}}$, calculate ${\textstyle {\frac {\partial (K-H)}{\partial P}}}$:

$${\begin{aligned}\left({\frac {\partial (K-H)}{\partial P}}\right)_{Q,P,t}={\frac {\partial K}{\partial P}}-{\frac {\partial H}{\partial p}}{\frac {\partial p}{\partial P}}-{\frac {\partial H}{\partial q}}{\frac {\partial q}{\partial P}}-{\frac {\partial H}{\partial t}}\left({\frac {\partial t}{\partial P}}\right)_{Q,P,t}={\dot {Q}}+{\dot {p}}{\frac {\partial q}{\partial P}}-{\dot {q}}{\frac {\partial p}{\partial P}}\\={\frac {\partial Q}{\partial t}}+{\frac {\partial Q}{\partial q}}\cdot {\dot {q}}+{\frac {\partial Q}{\partial p}}\cdot {\dot {p}}+{\dot {p}}{\frac {\partial q}{\partial P}}-{\dot {q}}{\frac {\partial p}{\partial P}}\\={\dot {q}}\left({\frac {\partial Q}{\partial q}}-{\frac {\partial p}{\partial P}}\right)+{\dot {p}}\left({\frac {\partial q}{\partial P}}+{\frac {\partial Q}{\partial p}}\right)+{\frac {\partial Q}{\partial t}}\end{aligned}}$$Hence: ${\textstyle {\frac {\partial (K-H)}{\partial P}}={\frac {\partial Q}{\partial t}}}$, if canonical transformation rules are applied.

Similarly:

$${\begin{aligned}\left({\frac {\partial (K-H)}{\partial Q}}\right)_{Q,P,t}={\frac {\partial K}{\partial Q}}-{\frac {\partial H}{\partial p}}{\frac {\partial p}{\partial Q}}-{\frac {\partial H}{\partial q}}{\frac {\partial q}{\partial Q}}-{\frac {\partial H}{\partial t}}\left({\frac {\partial t}{\partial Q}}\right)_{Q,P,t}=-{\dot {P}}+{\dot {p}}{\frac {\partial q}{\partial Q}}-{\dot {q}}{\frac {\partial p}{\partial Q}}\\=-{\frac {\partial P}{\partial t}}-{\frac {\partial P}{\partial q}}\cdot {\dot {q}}-{\frac {\partial P}{\partial p}}\cdot {\dot {p}}+{\dot {p}}{\frac {\partial q}{\partial Q}}-{\dot {q}}{\frac {\partial p}{\partial Q}}\\=-\left({\dot {q}}\left({\frac {\partial P}{\partial q}}+{\frac {\partial p}{\partial Q}}\right)+{\dot {p}}\left({\frac {\partial P}{\partial p}}-{\frac {\partial q}{\partial Q}}\right)+{\frac {\partial P}{\partial t}}\right)\end{aligned}}$$Hence: ${\textstyle {\frac {\partial (K-H)}{\partial Q}}=-{\frac {\partial P}{\partial t}}}$, if canonical transformation rules are applied.

The above two relations can be combined in matrix form as: ${\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}$ (which will also retain same form for extended canonical transformation) where we have used the result ${\textstyle {\frac {\partial G}{\partial t}}=K-H}$.

The canonical transformation relations can now be restated to include time dependance:$${\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}$$$${\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}$$

Symplectic Condition

From:$${\dot {\eta }}=J\nabla _{\eta }H=J(M^{T}\nabla _{\varepsilon }H)$$

Similarly we find:$${\dot {\varepsilon }}=M{\dot {\eta }}+{\frac {\partial \varepsilon }{\partial t}}=MJM^{T}\nabla _{\varepsilon }H+{\frac {\partial \varepsilon }{\partial t}}$$or:$${\dot {\varepsilon }}=J\nabla _{\varepsilon }K=J\nabla _{\varepsilon }H+J\nabla _{\varepsilon }\left({\frac {\partial G}{\partial t}}\right)$$Where the last terms of each equation cancel due to ${\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}$ condition from canonical transformations. Hence leaving the symplectic relation: ${\textstyle MJM^{T}=J}$. It follows from the above two equations that the symplectic condition implies the equation: ${\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}$, from which the indirect conditions can be recovered. Thus, symplectic conditions and indirect conditions can be said to be equivalent.

The results for invariance of Poisson and Lagrange brackets also follows from here. However, bilinear condition will remain in the domain of restricted canonical transformations only.

The initial analysis of canonical transformations is hence consistent with this generalization. We can also observe that since ${\textstyle {\frac {\partial (K-H)}{\partial P}}={\frac {\partial Q}{\partial t}}}$ and ${\textstyle {\frac {\partial (K-H)}{\partial Q}}=-{\frac {\partial P}{\partial t}}}$ if Q and P do not explicitly depend on time, ${\textstyle K=H+{\frac {\partial G}{\partial t}}(t)}$ can be taken.

Extended Canonical Transformation

Canonical transformation relations

By solving for:$$\lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt}}$$with various forms of generating function, we instead get the relation between K and H as ${\textstyle {\frac {\partial G}{\partial t}}=K-\lambda H}$ which also applies for ${\textstyle \lambda =1}$ case. All results presented below can also be obtained by replacing ${\textstyle q\rightarrow {\sqrt {\lambda }}q}$, ${\textstyle p\rightarrow {\sqrt {\lambda }}p}$ and ${\textstyle H\rightarrow {\lambda }H}$ from known solutions, since these transformations retain the form of Hamilton's equations.

Using same steps previously used in previous generalization, with ${\textstyle {\frac {\partial G}{\partial t}}=K-\lambda H}$ in the general case, and retaining the equation ${\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial g}{\partial t}}\right)={\frac {\partial \varepsilon }{\partial t}}}$, we get extended canonical transformation partial differential relations:$${\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\lambda \left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\lambda \left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}$$$${\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\lambda \left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\lambda \left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}$$

Symplectic condition

From: $${\dot {\eta }}=J\nabla _{\eta }H=J(M^{T}\nabla _{\varepsilon }H)$$Similarly we find:$${\dot {\varepsilon }}=M{\dot {\eta }}+{\frac {\partial \varepsilon }{\partial t}}=MJM^{T}\nabla _{\varepsilon }H+{\frac {\partial \varepsilon }{\partial t}}$$or using ${\textstyle {\frac {\partial G}{\partial t}}=K-\lambda H}$:$${\dot {\varepsilon }}=J\nabla _{\varepsilon }K=\lambda J\nabla _{\varepsilon }H+J\nabla _{\varepsilon }\left({\frac {\partial G}{\partial t}}\right)$$The second part of each equation cancels as usual. Hence the condition for extended canonical transformation instead becomes: ${\textstyle MJM^{T}=\lambda J}$

Poisson and Lagrange Brackets

For Poisson brackets:$$\{u,v\}_{\eta }=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)=(M^{T}\nabla _{\varepsilon }u)^{T}J(M^{T}\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}MJM^{T}(\nabla _{\varepsilon }v)=\lambda (\nabla _{\varepsilon }u)^{T}J(\nabla _{\varepsilon }v)=\lambda \{u,v\}_{\varepsilon }$$and for Lagrange brackets:

${\textstyle (MJM^{T})_{ij}=[\eta _{i},\eta _{j}]_{\varepsilon }=\lambda J_{ij}=\lambda [\eta _{i},\eta _{j}]_{\eta }\Rightarrow [u,v]_{\varepsilon }=\lambda [u,v]_{\eta }}$

Bilinear Invariance condition for restricted canonical transformation

A restricted canonical transformation can also have ${\textstyle \lambda \neq 1}$ but does not have bilinear invariance condition as in ${\textstyle \lambda =1}$ case. Instead it is modified to:$$\lambda \left(\sum \delta q\cdot dp-\delta p\cdot dq\right)=\sum \delta Q\cdot dP-\delta P\cdot dQ$$

Liouville's theorem

Liouville's theorem is changed into:$$\int \mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} ={\frac {1}{\lambda ^{N}}}\int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P}$$where N is the number of generalized coordinate or generalized momentum.

Motion as a canonical transformation

Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If $\mathbf {Q} (t)\equiv \mathbf {q} (t+\tau )$ and $\mathbf {P} (t)\equiv \mathbf {p} (t+\tau )$, then Hamilton's principle is automatically satisfied$$\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)\right]dt=\delta \int _{t_{1}+\tau }^{t_{2}+\tau }\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t+\tau )\right]dt=0$$since a valid trajectory $(\mathbf {q} (t),\mathbf {p} (t))$ should always satisfy Hamilton's principle, regardless of the endpoints.

Examples

• The translation $\mathbf {Q} (\mathbf {q} ,\mathbf {p} )=\mathbf {q} +\mathbf {a} ,\mathbf {P} (\mathbf {q} ,\mathbf {p} )=\mathbf {p} +\mathbf {b}$ where $\mathbf {a} ,\mathbf {b}$ are two constant vectors is a canonical transformation. Indeed, the Jacobian matrix is the identity, which is symplectic: $I^{\text{T}}JI=J$.
• Set $\mathbf {x} =(q,p)$ and $\mathbf {X} =(Q,P)$, the transformation $\mathbf {X} (\mathbf {x} )=R\mathbf {x}$ where $R\in SO(2)$ is a rotation matrix of order 2 is canonical. Keeping in mind that special orthogonal matrices obey $R^{\text{T}}R=I$ it's easy to see that the Jacobian is symplectic. Be aware that this example only works in dimension 2: $SO(2)$ is the only special orthogonal group in which every matrix is symplectic.
• 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 $${\frac {\partial X}{\partial x}}={\begin{bmatrix}1&f'(p)\\0&1\end{bmatrix}}$$ 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 $$\sum _{i}p_{i}\,dq^{i}$$ 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 q 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

• Goldstein, Herbert (1980). Classical mechanics (2d ed.). Reading, Mass.: Addison-Wesley Pub. Co. p. 380. ISBN 0-201-02918-9.
• Landau, L. D.; Lifshitz, E. M. (1975) [1939]. Mechanics. Translated by Bell, S. J.; Sykes, J. B. (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-7506-28969.