In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) 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 q → Q 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

$${\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q))_{i))}\ ,}$$

${\displaystyle \left\{\ (P_{1},Q_{1}),\ (P_{2},Q_{2}),\ (P_{3},Q_{3}),\ \ldots \ \right\))$

${\displaystyle P_{i))$

${\displaystyle Q_{i},}$

${\displaystyle i=1,2,\ldots \ N,}$

${\displaystyle N}$

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 q represent a list of N generalized coordinates that need not transform like a vector under rotation and similarly p represents the corresponding generalized momentum, e.g.,

$${\displaystyle {\begin{aligned}\mathbf {q} &\equiv \left(q_{1},q_{2},\ldots ,q_{N-1},q_{N}\right)\\\mathbf {p} &\equiv \left(p_{1},p_{2},\ldots ,p_{N-1},p_{N}\right).\end{aligned))}$$

A dot over a variable or list signifies the time derivative, e.g.,

$${\displaystyle {\dot {\mathbf {q} ))\equiv {\frac {d\mathbf {q} }{dt))}$$

$${\displaystyle {\dot {\mathbf {p} ))=-{\frac {\partial f}{\partial \mathbf {q} ))\quad \Longleftrightarrow \quad {\dot {p_{i))}=-{\frac {\partial f}{\partial {q_{i))))\quad (\forall i).}$$

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.,

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

The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value. The coordinates after transformation are similarly labelled with Q for transformed generalized coordinates and P for transformed generalized momentum.

Conditions for restricted canonical transformation

Restricted canonical transformations are coordinate transformations where transformed coordinates Q and P do not have explicit time dependance, ie. ${\textstyle \mathbf {Q} =\mathbf {Q} (\mathbf {q} ,\mathbf {p} )}$ and ${\textstyle \mathbf {P} =\mathbf {P} (\mathbf {q} ,\mathbf {p} )}$. The following conditions can be generalized to canonical transformation with the exception of bilinear invariance condition which is only applicable under restricted canonical transformations.

Indirect conditions

The functional form of Hamilton's equations is

$${\displaystyle {\begin{aligned}{\dot {\mathbf {p} ))&=-{\frac {\partial H}{\partial \mathbf {q} ))\\{\dot {\mathbf {q} ))&={\frac {\partial H}{\partial \mathbf {p} ))\end{aligned))}$$

$${\displaystyle K(\mathbf {Q} ,\mathbf {P} ,t)=H(q(\mathbf {Q} ,\mathbf {P} ),p(\mathbf {Q} ,\mathbf {P} ),t)+{\frac {\partial G}{\partial t))(t).}$$

$${\displaystyle {\begin{aligned}\left({\frac {\partial H}{\partial q))\right)_{\mathbf {q} ,\mathbf {p} ,t}=\left({\frac {\partial K}{\partial q))\right)_{\mathbf {q} ,\mathbf {p} ,t}\\\left({\frac {\partial H}{\partial p))\right)_{\mathbf {q} ,\mathbf {p} ,t}=\left({\frac {\partial K}{\partial p))\right)_{\mathbf {q} ,\mathbf {p} ,t}\end{aligned))}$$

By definition, the transformed coordinates have analogous dynamics

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

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

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

$${\displaystyle {\begin{aligned}{\dot {Q))_{m}&={\frac {\partial Q_{m)){\partial \mathbf {q} ))\cdot {\dot {\mathbf {q} ))+{\frac {\partial Q_{m)){\partial \mathbf {p} ))\cdot {\dot {\mathbf {p} ))\\&={\frac {\partial Q_{m)){\partial \mathbf {q} ))\cdot {\frac {\partial H}{\partial \mathbf {p} ))-{\frac {\partial Q_{m)){\partial \mathbf {p} ))\cdot {\frac {\partial H}{\partial \mathbf {q} ))\\&=\lbrace Q_{m},H\rbrace \end{aligned))}$$

We also have the identity for the conjugate momentum P_m

$${\displaystyle {\begin{aligned}&{\frac {\partial K(\mathbf {Q} ,\mathbf {P} ,t)}{\partial P_{m))}\\&={\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {q} ))\cdot {\frac {\partial \mathbf {q} }{\partial P_{m))}+{\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {p} ))\cdot {\frac {\partial \mathbf {p} }{\partial P_{m))}\\&={\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {q} ))\cdot {\frac {\partial \mathbf {q} }{\partial P_{m))}+{\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {p} ))\cdot {\frac {\partial \mathbf {p} }{\partial P_{m))}\\&={\frac {\partial H}{\partial \mathbf {q} ))\cdot {\frac {\partial \mathbf {q} }{\partial P_{m))}+{\frac {\partial H}{\partial \mathbf {p} ))\cdot {\frac {\partial \mathbf {p} }{\partial P_{m))}\end{aligned))}$$

If the transformation is canonical, these two must be equal, resulting in the equations

$${\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m)){\partial p_{n))}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial q_{n)){\partial P_{m))}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial Q_{m)){\partial q_{n))}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial p_{n)){\partial P_{m))}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned))}$$

The analogous argument for the generalized momenta P_m leads to two other sets of equations

$${\displaystyle {\begin{aligned}\left({\frac {\partial P_{m)){\partial p_{n))}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial q_{n)){\partial Q_{m))}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial P_{m)){\partial q_{n))}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial p_{n)){\partial Q_{m))}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned))}$$

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

Symplectic condition

Sometimes the Hamiltonian relations are represented as:

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

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 ${\displaystyle {\dot {\eta ))=J\nabla _{\eta }H}$ relation in terms of partial derivatives with new variables:

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

Similarly we find:

$${\displaystyle {\dot {\varepsilon ))=M{\dot {\eta ))=MJM^{T}\nabla _{\varepsilon }H}$$

${\textstyle \nabla _{\varepsilon }K=\nabla _{\varepsilon }H}$

$${\displaystyle {\dot {\varepsilon ))=J\nabla _{\varepsilon }K=J\nabla _{\varepsilon }H}$$

we get the symplectic condition:^[2]

$${\displaystyle MJM^{T}=J}$$

${\displaystyle \varepsilon }$

${\textstyle {\mathcal {P))(\varepsilon )=MJM^{T))$

${\displaystyle \eta }$

${\textstyle {\mathcal {L))(\eta )=M^{T}JM}$

${\textstyle M^{T}JM=J}$

${\textstyle J^{-1}=-J}$

${\textstyle M}$

Invariance of Poisson Bracket

The Poisson bracket which is defined as:

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

$${\displaystyle \{u,v\}_{\eta }:=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)}$$

$${\displaystyle \{u,v\}_{\eta }=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)=(M^{T}\nabla _{\varepsilon }u)^{T}J(M^{T}\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}MJM^{T}(\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}J(\nabla _{\varepsilon }v)=\{u,v\}_{\varepsilon ))$$

The symplectic condition can also be recovered by taking ${\textstyle u=\varepsilon _{i))$ and ${\textstyle v=\varepsilon _{j))$ which shows that ${\textstyle (MJM^{T})_{ij}=J_{ij))$. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that ${\textstyle {\mathcal {P))_{ij}(\varepsilon )=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=(MJM^{T})_{ij))$, which is also the result of explicitly calculating the matrix element by expanding it.^[3]

Invariance of Lagrange Bracket

The Lagrange bracket which is defined as:

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

can be represented in matrix form as:

$${\displaystyle [u,v]_{\eta }:=\left({\frac {\partial \eta }{\partial u))\right)^{T}J\left({\frac {\partial \eta }{\partial v))\right)}$$

Using similar derivation, we get:

$${\displaystyle [u,v]_{\varepsilon }=(\partial _{u}\varepsilon )^{T}\,J\,(\partial _{v}\varepsilon )=(M\,\partial _{u}\eta )^{T}\,J\,(M\,\partial _{v}\eta )=(\partial _{u}\eta )^{T}\,M^{T}JM\,(\partial _{v}\eta )=(\partial _{u}\eta )^{T}\,J\,(\partial _{v}\eta )=[u,v]_{\eta ))$$

${\textstyle u=\eta _{i))$

${\textstyle v=\eta _{j))$

${\textstyle (M^{T}JM)_{ij}=J_{ij))$

${\textstyle {\mathcal {L))_{ij}(\eta )=[\eta _{i},\eta _{j}]_{\varepsilon }=(M^{T}JM)_{ij))$

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:^[5]

${\textstyle d\varepsilon =(dq_{1},dp_{1},0,0,\ldots ),\quad \delta \varepsilon =(\delta q_{1},\delta p_{1},0,0,\ldots ).}$

The area of the infinitesimal parallelogram is given by:

${\textstyle \delta a(12)=dq_{1}\delta p_{1}-\delta q_{1}dp_{1}={(\delta \varepsilon )}^{T}\,J\,d\varepsilon .}$

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

${\textstyle \delta a(12)={(\delta \varepsilon )}^{T}\,J\,d\varepsilon ={(M\delta \eta )}^{T}\,J\,Md\eta ={(\delta \eta )}^{T}\,M^{T}JM\,d\eta ={(\delta \eta )}^{T}\,J\,d\eta =\delta A(12).}$

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 ${\textstyle {(d\varepsilon )}^{T}\,J\,\delta \varepsilon }$ under canonical transformation, expanded as:

$${\displaystyle \sum \delta q\cdot dp-\delta p\cdot dq=\sum \delta Q\cdot dP-\delta P\cdot dQ}$$

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.,

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

By calculus, the latter integral must equal the former times the determinant of Jacobian M

$${\displaystyle \int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P} =\int \det(M)\,\mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} }$$

${\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )))}$

Exploiting the "division" property of Jacobians yields

$${\displaystyle M\equiv {\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {P} )))\left/{\frac {\partial (\mathbf {q} ,\mathbf {p} )}{\partial (\mathbf {q} ,\mathbf {P} )))\right.}$$

Eliminating the repeated variables gives

$${\displaystyle M\equiv {\frac {\partial (\mathbf {Q} )}{\partial (\mathbf {q} )))\left/{\frac {\partial (\mathbf {p} )}{\partial (\mathbf {P} )))\right.}$$

Application of the indirect conditions above yields ${\displaystyle \operatorname {det} (M)=1}$.^[8]

Generating function approach

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 ${\displaystyle {\mathcal {L))_{qp}=\mathbf {p} \cdot {\dot {\mathbf {q} ))-H(\mathbf {q} ,\mathbf {p} ,t)}$ and ${\displaystyle {\mathcal {L))_{QP}=\mathbf {P} \cdot {\dot {\mathbf {Q} ))-K(\mathbf {Q} ,\mathbf {P} ,t)}$ 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):

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

One way for both variational integral equalities to be satisfied is to have

$${\displaystyle \lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} ))-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} ))-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt))}$$

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^[9]

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

Type 1 generating function

The type 1 generating function G₁ depends only on the old and new generalized coordinates

$${\displaystyle G\equiv G_{1}(\mathbf {q} ,\mathbf {Q} ,t)}$$

$${\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} ))-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} ))-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{1)){\partial t))+{\frac {\partial G_{1)){\partial \mathbf {q} ))\cdot {\dot {\mathbf {q} ))+{\frac {\partial G_{1)){\partial \mathbf {Q} ))\cdot {\dot {\mathbf {Q} ))}$$

Since the new and old coordinates are each independent, the following 2N + 1 equations must hold

$${\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{1)){\partial \mathbf {q} ))\\\mathbf {P} &=-{\frac {\partial G_{1)){\partial \mathbf {Q} ))\\K&=H+{\frac {\partial G_{1)){\partial t))\end{aligned))}$$

These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations

$${\displaystyle \ \mathbf {p} ={\frac {\ \partial G_{1}\ }{\partial \mathbf {q} ))\ }$$

$${\displaystyle \mathbf {P} =-{\frac {\partial G_{1)){\partial \mathbf {Q} ))}$$

$${\displaystyle K=H+{\frac {\partial G_{1)){\partial t))}$$

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let

$${\displaystyle G_{1}\equiv \mathbf {q} \cdot \mathbf {Q} }$$

$${\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{1)){\partial \mathbf {q} ))=\mathbf {Q} \\\mathbf {P} &=-{\frac {\partial G_{1)){\partial \mathbf {Q} ))=-\mathbf {q} \end{aligned))}$$

Type 2 generating function

The type 2 generating function ${\displaystyle G_{2}(\mathbf {q} ,\mathbf {P} ,t)}$ depends only on the old generalized coordinates and the new generalized momenta

$${\displaystyle G\equiv G_{2}(\mathbf {q} ,\mathbf {P} ,t)-\mathbf {Q} \cdot \mathbf {P} }$$

${\displaystyle -\mathbf {Q} \cdot \mathbf {P} }$

$${\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} ))-H(\mathbf {q} ,\mathbf {p} ,t)=-\mathbf {Q} \cdot {\dot {\mathbf {P} ))-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{2)){\partial t))+{\frac {\partial G_{2)){\partial \mathbf {q} ))\cdot {\dot {\mathbf {q} ))+{\frac {\partial G_{2)){\partial \mathbf {P} ))\cdot {\dot {\mathbf {P} ))}$$

Since the old coordinates and new momenta are each independent, the following 2N + 1 equations must hold

$${\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{2)){\partial \mathbf {q} ))\\\mathbf {Q} &={\frac {\partial G_{2)){\partial \mathbf {P} ))\\K&=H+{\frac {\partial G_{2)){\partial t))\end{aligned))}$$

These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations

$${\displaystyle \mathbf {p} ={\frac {\partial G_{2)){\partial \mathbf {q} ))}$$

$${\displaystyle \mathbf {Q} ={\frac {\partial G_{2)){\partial \mathbf {P} ))}$$

$${\displaystyle K=H+{\frac {\partial G_{2)){\partial t))}$$

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let

$${\displaystyle G_{2}\equiv \mathbf {g} (\mathbf {q} ;t)\cdot \mathbf {P} }$$

$${\displaystyle \mathbf {Q} ={\frac {\partial G_{2)){\partial \mathbf {P} ))=\mathbf {g} (\mathbf {q} ;t)}$$

Type 3 generating function

The type 3 generating function ${\displaystyle G_{3}(\mathbf {p} ,\mathbf {Q} ,t)}$ depends only on the old generalized momenta and the new generalized coordinates

$${\displaystyle G\equiv G_{3}(\mathbf {p} ,\mathbf {Q} ,t)+\mathbf {q} \cdot \mathbf {p} }$$

${\displaystyle \mathbf {q} \cdot \mathbf {p} }$

$${\displaystyle -\mathbf {q} \cdot {\dot {\mathbf {p} ))-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} ))-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{3)){\partial t))+{\frac {\partial G_{3)){\partial \mathbf {p} ))\cdot {\dot {\mathbf {p} ))+{\frac {\partial G_{3)){\partial \mathbf {Q} ))\cdot {\dot {\mathbf {Q} ))}$$

Since the new and old coordinates are each independent, the following 2N + 1 equations must hold

$${\displaystyle {\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{3)){\partial \mathbf {p} ))\\\mathbf {P} &=-{\frac {\partial G_{3)){\partial \mathbf {Q} ))\\K&=H+{\frac {\partial G_{3)){\partial t))\end{aligned))}$$

These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations

$${\displaystyle \mathbf {q} =-{\frac {\partial G_{3)){\partial \mathbf {p} ))}$$

$${\displaystyle \mathbf {P} =-{\frac {\partial G_{3)){\partial \mathbf {Q} ))}$$

$${\displaystyle K=H+{\frac {\partial G_{3)){\partial t))}$$

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 ${\displaystyle G_{4}(\mathbf {p} ,\mathbf {P} ,t)}$ depends only on the old and new generalized momenta

$${\displaystyle G\equiv G_{4}(\mathbf {p} ,\mathbf {P} ,t)+\mathbf {q} \cdot \mathbf {p} -\mathbf {Q} \cdot \mathbf {P} }$$

${\displaystyle \mathbf {q} \cdot \mathbf {p} -\mathbf {Q} \cdot \mathbf {P} }$

$${\displaystyle -\mathbf {q} \cdot {\dot {\mathbf {p} ))-H(\mathbf {q} ,\mathbf {p} ,t)=-\mathbf {Q} \cdot {\dot {\mathbf {P} ))-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{4)){\partial t))+{\frac {\partial G_{4)){\partial \mathbf {p} ))\cdot {\dot {\mathbf {p} ))+{\frac {\partial G_{4)){\partial \mathbf {P} ))\cdot {\dot {\mathbf {P} ))}$$

Since the new and old coordinates are each independent, the following 2N + 1 equations must hold

$${\displaystyle {\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{4)){\partial \mathbf {p} ))\\\mathbf {Q} &={\frac {\partial G_{4)){\partial \mathbf {P} ))\\K&=H+{\frac {\partial G_{4)){\partial t))\end{aligned))}$$

These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations

$${\displaystyle \mathbf {q} =-{\frac {\partial G_{4)){\partial \mathbf {p} ))}$$

$${\displaystyle \mathbf {Q} ={\frac {\partial G_{4)){\partial \mathbf {P} ))}$$

$${\displaystyle K=H+{\frac {\partial G_{4)){\partial t))}$$

Restrictions on generating functions

For example, using generating function of second kind: ${\textstyle {p}_{i}={\frac {\partial G_{2)){\partial {q}_{i))))$ and ${\textstyle {Q}_{i}={\frac {\partial G_{2)){\partial {P}_{i))))$, the first set of equations consisting of variables ${\textstyle \mathbf {p} }$, ${\textstyle \mathbf {q} }$ and ${\textstyle \mathbf {P} }$ has to be inverted to get ${\textstyle \mathbf {P} (\mathbf {q} ,\mathbf {p} )}$. This process is possible when the matrix defined by ${\textstyle a_{ij}={\frac {\partial {p}_{i}(\mathbf {q} ,\mathbf {P} )}{\partial P_{j))))$ is non-singular.^[10]

$${\displaystyle \left|{\begin{array}{l l l}{\displaystyle {\frac {\partial ^{2}G_{2)){\partial P_{1}\partial q_{1))))&{\cdots }&{\displaystyle {\frac {\partial ^{2}G_{2)){\partial P_{1}\partial q_{n))))\\{\quad \vdots }&{\ddots }&{\quad \vdots }\\{\displaystyle {\frac {\partial ^{2}G_{2)){\partial P_{n}\partial q_{1))))&{\cdots }&{\displaystyle {\frac {\partial ^{2}G_{2)){\partial P_{n}\partial q_{n))))\end{array))\right|{\neq 0))$$

Hence, restrictions are placed on generating functions to have the matrices: ${\textstyle \left[{\frac {\partial ^{2}G_{1)){\partial Q_{j}\partial q_{i))}\right]}$, ${\textstyle \left[{\frac {\partial ^{2}G_{2)){\partial P_{j}\partial q_{i))}\right]}$, ${\textstyle \left[{\frac {\partial ^{2}G_{3)){\partial p_{j}\partial Q_{i))}\right]}$ and ${\textstyle \left[{\frac {\partial ^{2}G_{4)){\partial p_{j}\partial P_{i))}\right]}$, being non-singular.^[11][12]

Limitations of generating functions

Since ${\textstyle \left[{\frac {\partial ^{2}G_{2)){\partial P_{j}\partial q_{i))}\right]}$ is non-singular, it implies that ${\textstyle \left[{\frac {\partial {p}_{i}(\mathbf {q} ,\mathbf {P} )}{\partial P_{j))}\right]}$ is also non-singular. Since the matrix ${\textstyle \left[{\frac {\partial P_{i}(\mathbf {q} ,\mathbf {p} )}{\partial p_{j))}\right]}$ is inverse of ${\textstyle \left[{\frac {\partial {p}_{i}(\mathbf {q} ,\mathbf {P} )}{\partial P_{j))}\right]}$, the transformations of type 2 generating functions always have a non-singular ${\textstyle \left[{\frac {\partial P_{i}(\mathbf {q} ,\mathbf {p} )}{\partial p_{j))}\right]}$ matrix. Similarly, it can be stated that type 1 and type 4 generating functions always have a non-singular ${\textstyle \left[{\frac {\partial Q_{i}(\mathbf {q} ,\mathbf {p} )}{\partial p_{j))}\right]}$ matrix whereas type 2 and type 3 generating functions always have a non-singular ${\textstyle \left[{\frac {\partial P_{i}(\mathbf {q} ,\mathbf {p} )}{\partial p_{j))}\right]}$ matrix. Hence, the canonical transformations resulting from these generating functions are not completely general.^[13]

In other words, since (Q, P) and (q, p) are each 2N independent functions, it follows that to have generating function of the form ${\textstyle G_{1}(\mathbf {q} ,\mathbf {Q} ,t)}$ and ${\displaystyle G_{4}(\mathbf {p} ,\mathbf {P} ,t)}$ or${\displaystyle G_{2}(\mathbf {q} ,\mathbf {P} ,t)}$ and ${\displaystyle G_{3}(\mathbf {p} ,\mathbf {Q} ,t)}$, the corresponding Jacobian matrices ${\textstyle \left[{\frac {\partial Q_{i)){\partial p_{j))}\right]}$ and ${\textstyle \left[{\frac {\partial P_{i)){\partial p_{j))}\right]}$ are restricted to be non singular, ensuring that the generating function is a function of 2N + 1 independent variables. However, as a feature of canonical transformations, it is always possible to choose 2N such independent functions from sets (q, p) or (Q, P), 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.^[14]

Canonical transformation conditions

Canonical transformation relations

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

$${\displaystyle {\begin{aligned}\left({\frac {\partial (K-H)}{\partial P))\right)_{Q,P,t}={\frac {\partial K}{\partial P))-{\frac {\partial H}{\partial p)){\frac {\partial p}{\partial P))-{\frac {\partial H}{\partial q)){\frac {\partial q}{\partial P))-{\frac {\partial H}{\partial t))\left({\frac {\partial t}{\partial P))\right)_{Q,P,t}={\dot {Q))+{\dot {p)){\frac {\partial q}{\partial P))-{\dot {q)){\frac {\partial p}{\partial P))\\={\frac {\partial Q}{\partial t))+{\frac {\partial Q}{\partial q))\cdot {\dot {q))+{\frac {\partial Q}{\partial p))\cdot {\dot {p))+{\dot {p)){\frac {\partial q}{\partial P))-{\dot {q)){\frac {\partial p}{\partial P))\\={\dot {q))\left({\frac {\partial Q}{\partial q))-{\frac {\partial p}{\partial P))\right)+{\dot {p))\left({\frac {\partial q}{\partial P))+{\frac {\partial Q}{\partial p))\right)+{\frac {\partial Q}{\partial t))\end{aligned))}$$

${\textstyle {\frac {\partial (K-H)}{\partial P))={\frac {\partial }{\partial P))\left({\frac {\partial G}{\partial t))\right){\bigg |}_{Q,P,t))$

${\textstyle {\dot {q))}$

${\textstyle {\dot {p))}$

${\textstyle {\frac {\partial (K-H)}{\partial P))={\frac {\partial Q}{\partial t))}$

Similarly:

$${\displaystyle {\begin{aligned}\left({\frac {\partial (K-H)}{\partial Q))\right)_{Q,P,t}={\frac {\partial K}{\partial Q))-{\frac {\partial H}{\partial p)){\frac {\partial p}{\partial Q))-{\frac {\partial H}{\partial q)){\frac {\partial q}{\partial Q))-{\frac {\partial H}{\partial t))\left({\frac {\partial t}{\partial Q))\right)_{Q,P,t}=-{\dot {P))+{\dot {p)){\frac {\partial q}{\partial Q))-{\dot {q)){\frac {\partial p}{\partial Q))\\=-{\frac {\partial P}{\partial t))-{\frac {\partial P}{\partial q))\cdot {\dot {q))-{\frac {\partial P}{\partial p))\cdot {\dot {p))+{\dot {p)){\frac {\partial q}{\partial Q))-{\dot {q)){\frac {\partial p}{\partial Q))\\=-\left({\dot {q))\left({\frac {\partial P}{\partial q))+{\frac {\partial p}{\partial Q))\right)+{\dot {p))\left({\frac {\partial P}{\partial p))-{\frac {\partial q}{\partial Q))\right)+{\frac {\partial P}{\partial t))\right)\end{aligned))}$$

${\textstyle {\frac {\partial (K-H)}{\partial Q))=-{\frac {\partial P}{\partial t))}$

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 are hence said to be equivalent to ${\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t))\right)={\frac {\partial \varepsilon }{\partial t))}$ in this context.

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

$${\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m)){\partial p_{n))}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\left({\frac {\partial q_{n)){\partial P_{m))}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial Q_{m)){\partial q_{n))}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\left({\frac {\partial p_{n)){\partial P_{m))}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned))}$$

$${\displaystyle {\begin{aligned}\left({\frac {\partial P_{m)){\partial p_{n))}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\left({\frac {\partial q_{n)){\partial Q_{m))}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial P_{m)){\partial q_{n))}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\left({\frac {\partial p_{n)){\partial Q_{m))}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned))}$$

${\textstyle {\frac {\partial (K-H)}{\partial P))={\frac {\partial Q}{\partial t))}$

${\textstyle {\frac {\partial (K-H)}{\partial Q))=-{\frac {\partial P}{\partial t))}$

${\textstyle K=H+{\frac {\partial G}{\partial t))(t)}$

Symplectic Condition

From:

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

Similarly we find:

$${\displaystyle {\dot {\varepsilon ))=M{\dot {\eta ))+{\frac {\partial \varepsilon }{\partial t))=MJM^{T}\nabla _{\varepsilon }H+{\frac {\partial \varepsilon }{\partial t))}$$

$${\displaystyle {\dot {\varepsilon ))=J\nabla _{\varepsilon }K=J\nabla _{\varepsilon }H+J\nabla _{\varepsilon }\left({\frac {\partial G}{\partial t))\right)}$$

${\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t))\right)={\frac {\partial \varepsilon }{\partial t))}$

${\textstyle MJM^{T}=J}$

${\textstyle M^{T}JM=J}$

${\textstyle J\left(\nabla _{\varepsilon }{\frac {\partial G}{\partial t))\right)={\frac {\partial \varepsilon }{\partial t))}$

Invariance of Poisson and Lagrange Bracket

Since ${\textstyle {\mathcal {P))_{ij}(\varepsilon )=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=(MJM^{T})_{ij}=J_{ij))$ and ${\textstyle {\mathcal {L))_{ij}(\eta )=[\eta _{i},\eta _{j}]_{\varepsilon }=(M^{T}JM)_{ij}=J_{ij))$ where the symplectic condition is used in the last equalities. Using ${\textstyle \{\varepsilon _{i},\varepsilon _{j}\}_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta }=J_{ij))$, the equalities ${\textstyle \{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=\{\varepsilon _{i},\varepsilon _{j}\}_{\varepsilon ))$ and ${\textstyle [\eta _{i},\eta _{j}]_{\varepsilon }=[\eta _{i},\eta _{j}]_{\eta ))$ are obtained which imply the invariance of Poisson and Lagrange brackets.

Extended Canonical Transformation

Canonical transformation relations

By solving for:

$${\displaystyle \lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} ))-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} ))-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt))}$$

${\textstyle {\frac {\partial G}{\partial t))=K-\lambda H}$

${\textstyle \lambda =1}$

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 it retains the form of Hamilton's equations. The extended canonical transformations are hence said to be result of a canonical transformation (${\textstyle \lambda =1}$) and a trivial canonical transformation (${\textstyle \lambda \neq 1}$) which has ${\textstyle MJM^{T}=\lambda J}$ (for the given example, ${\textstyle M={\sqrt {\lambda ))I}$ which satisfies the condition).^[15]

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: