Editing Canonical transformation (section)

==Generating function approach==
{{main|Generating function (physics)}}
To ''guarantee'' a valid transformation between {{math|('''q''', '''p''', ''H'')}} and {{math|('''Q''', '''P''', ''K'')}}, we may resort to a direct '''generating function''' approach. Both sets of variables must obey [[action (physics)|Hamilton's principle]]. That is the [[action integral]] over the [[Lagrangian mechanics|Lagrangians]] <math>\mathcal{L}_{qp}=\mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t)</math> and <math>\mathcal{L}_{QP}=\mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t)</math>, 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 [[Hamilton equations#Deriving Hamilton's equations|here]]):

<math display="block">\begin{align}
\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{align}</math>

One way for both [[calculus of variations|variational integral]] equalities to be satisfied is to have

<math display="block">\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} </math>

Lagrangians are not unique: one can always multiply by a constant {{mvar|λ}} and add a total time derivative {{math|{{sfrac|''dG''|''dt''}}}} and yield the same equations of motion (as [[b:Classical Mechanics/Lagrange Theory#Is the Lagrangian unique?|discussed on Wikibooks]]). In general, the scaling factor {{mvar|λ}} is set equal to one; canonical transformations for which {{math|''λ'' ≠ 1}} are called '''extended canonical transformations'''. {{math|{{sfrac|''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 {{mvar|G}} is a [[generating function (physics)|generating function]] of one old [[canonical coordinates|canonical coordinate]] ({{math|'''q'''}} or {{math|'''p'''}}), one new [[canonical coordinates|canonical coordinate]] ({{math|'''Q'''}} or {{math|'''P'''}}) and (possibly) the time {{mvar|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 {{math|('''q''', '''p''') → ('''Q''', '''P''')}} is guaranteed to be canonical.

The various generating functions and its properties tabulated below is discussed in detail:
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+Properties of four basic canonical transformations<ref>{{harvnb|Goldstein|Poole|Safko|2007|p=373}}</ref>
!Generating function
! colspan="2" |Generating function derivatives
!Transformed Hamiltonian
! colspan="3" |Trivial cases
|-
|<math>G = G_1(q,Q,t) </math>
|<math>p = \frac{\partial G_1}{\partial q} </math>
|<math>P = - \frac{\partial G_1}{\partial Q} </math>
| rowspan="4" style="text-align: center;" |<math display="inline">K = H + \frac{\partial G}{\partial t} </math>
|<math>G_1 = qQ </math>
|<math>Q = p </math>
|<math>P = -q  </math>
|-
|<math>G = G_2(q,P,t) - QP   </math>
|<math>p = \frac{\partial G_2}{\partial q} </math>
|<math>Q = \frac{\partial G_2}{\partial P} </math>
|<math>G_2 = qP </math>
|<math>Q = q </math>
|<math>P = p </math>
|-
|<math>G = G_3(p,Q,t) + qp </math>
|<math>q = -\frac{\partial G_3}{\partial p}  </math>
|<math>P = -\frac{\partial G_3}{\partial Q}   </math>
|<math>G_3 = pQ </math>
|<math>Q = -q </math>
|<math>P = -p </math>
|-
|<math>G = G_4(p,P,t) + qp - QP </math>
|<math>q = -\frac{\partial G_4}{\partial p} </math>
|<math>Q = \frac{\partial G_4}{\partial P}    </math>
|<math>G_4 = pP </math>
|<math>Q = p </math>
|<math>P = -q  </math>
|}

===Type 1 generating function===
The type 1 generating function {{math|''G''<sub>1</sub>}} depends only on the old and new generalized coordinates <math display="inline">G \equiv G_{1}(\mathbf{q}, \mathbf{Q}, t)</math>. To derive the implicit transformation, we expand the defining equation above
<math display="block"> \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}}</math>

Since the new and old coordinates are each independent, the following {{math|2''N'' + 1}} equations must hold

<math display="block">\begin{align}
\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{align}</math>

These equations define the transformation {{math|('''q''', '''p''') → ('''Q''', '''P''')}} as follows: The ''first'' set of {{mvar|N}} equations <math display="inline">\ \mathbf{p} = \frac{\ \partial G_{1}\ }{ \partial \mathbf{q} }\ </math> define relations between the new [[generalized coordinates]] {{math|'''Q'''}} and the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. Ideally, one can invert these relations to obtain formulae for each {{math|''Q<sub>k</sub>''}} as a function of the old canonical coordinates. Substitution of these formulae for the {{math|'''Q'''}} coordinates into the ''second'' set of {{mvar|N}} equations <math display="inline">\mathbf{P} = -\frac{\partial G_{1}}{\partial \mathbf{Q}}</math> yields analogous formulae for the new generalized momenta {{math|'''P'''}} in terms of the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. We then invert both sets of formulae to obtain the ''old'' [[canonical coordinates]] {{math|('''q''', '''p''')}} as functions of the ''new'' [[canonical coordinates]] {{math|('''Q''', '''P''')}}. Substitution of the inverted formulae into the final equation <math display="inline">K = H + \frac{\partial G_{1}}{\partial t}</math> yields a formula for {{mvar|K}} as a function of the new [[canonical coordinates]] {{math|('''Q''', '''P''')}}.

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let <math display="inline">G_{1} \equiv \mathbf{q} \cdot \mathbf{Q}</math>. This results in swapping the generalized coordinates for the momenta and vice versa 

<math display="block">\begin{align}
\mathbf{p} &= \frac{\partial G_{1}}{\partial \mathbf{q}} = \mathbf{Q} \\
\mathbf{P} &= -\frac{\partial G_{1}}{\partial \mathbf{Q}} = -\mathbf{q}
\end{align}</math>

and {{math|1=''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 <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> depends only on the old [[generalized coordinates]] and the new generalized momenta <math display="inline">G \equiv  G_{2}(\mathbf{q}, \mathbf{P}, t)-\mathbf{Q} \cdot \mathbf{P}</math> where the <math>-\mathbf{Q} \cdot \mathbf{P}</math> 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

<math display="block"> \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}}</math>

Since the old coordinates and new momenta are each independent, the following {{math|2''N'' + 1}} equations must hold

<math display="block">\begin{align}
\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{align}</math>

These equations define the transformation {{math|('''q''', '''p''') → ('''Q''', '''P''')}} as follows: The ''first'' set of {{mvar|N}} equations <math display="inline">\mathbf{p} = \frac{\partial G_{2}}{\partial \mathbf{q}}</math> define relations between the new generalized momenta {{math|'''P'''}} and the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. Ideally, one can invert these relations to obtain formulae for each {{math|''P<sub>k</sub>''}} as a function of the old canonical coordinates. Substitution of these formulae for the {{math|'''P'''}} coordinates into the ''second'' set of {{mvar|N}} equations <math display="inline">\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}}</math> yields analogous formulae for the new generalized coordinates {{math|'''Q'''}} in terms of the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. We then invert both sets of formulae to obtain the ''old'' [[canonical coordinates]] {{math|('''q''', '''p''')}} as functions of the ''new'' [[canonical coordinates]] {{math|('''Q''', '''P''')}}. Substitution of the inverted formulae into the final equation <math display="inline">K = H + \frac{\partial G_{2}}{\partial t}</math> yields a formula for {{mvar|K}} as a function of the new [[canonical coordinates]] {{math|('''Q''', '''P''')}}.

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let <math display="inline">G_{2} \equiv \mathbf{g}(\mathbf{q}; t) \cdot \mathbf{P}</math> where {{math|'''g'''}} is a set of {{mvar|N}} functions. This results in a point transformation of the generalized coordinates <math display="inline">\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}} = \mathbf{g}(\mathbf{q}; t)</math>.

===Type 3 generating function===
The type 3 generating function <math>G_{3}(\mathbf{p}, \mathbf{Q}, t)</math> depends only on the old generalized momenta and the new generalized coordinates <math display="inline">G \equiv G_{3}(\mathbf{p}, \mathbf{Q}, t)+ \mathbf{q} \cdot \mathbf{p}</math> where the <math>\mathbf{q} \cdot \mathbf{p}</math> 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
<math display="block">-\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}}</math>

Since the new and old coordinates are each independent, the following {{math|2''N'' + 1}} equations must hold

<math display="block">\begin{align}
\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{align}</math>

These equations define the transformation {{math|('''q''', '''p''') → ('''Q''', '''P''')}} as follows: The ''first'' set of {{mvar|N}} equations <math display="inline"> \mathbf{q} = -\frac{\partial G_{3}}{\partial \mathbf{p}}</math> define relations between the new [[generalized coordinates]] {{math|'''Q'''}} and the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. Ideally, one can invert these relations to obtain formulae for each {{math|''Q<sub>k</sub>''}} as a function of the old canonical coordinates. Substitution of these formulae for the {{math|'''Q'''}} coordinates into the ''second'' set of {{mvar|N}} equations <math display="inline">\mathbf{P} = -\frac{\partial G_{3}}{\partial \mathbf{Q}}</math> yields analogous formulae for the new generalized momenta {{math|'''P'''}} in terms of the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. We then invert both sets of formulae to obtain the ''old'' [[canonical coordinates]] {{math|('''q''', '''p''')}} as functions of the ''new'' [[canonical coordinates]] {{math|('''Q''', '''P''')}}. Substitution of the inverted formulae into the final equation <math display="inline">K = H + \frac{\partial G_{3}}{\partial t}</math> yields a formula for {{mvar|K}} as a function of the new [[canonical coordinates]] {{math|('''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 <math>G_{4}(\mathbf{p}, \mathbf{P}, t)</math> depends only on the old and new generalized momenta <math display="inline">G \equiv G_{4}(\mathbf{p}, \mathbf{P}, t) +\mathbf{q} \cdot \mathbf{p} - \mathbf{Q} \cdot \mathbf{P} </math> where the <math>\mathbf{q} \cdot \mathbf{p} - \mathbf{Q} \cdot \mathbf{P}</math> terms represent a [[Legendre transformation]] to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above

<math display="block">-\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}} </math>

Since the new and old coordinates are each independent, the following {{math|2''N'' + 1}} equations must hold

<math display="block">\begin{align}
\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{align}</math>

These equations define the transformation {{math|('''q''', '''p''') → ('''Q''', '''P''')}} as follows: The ''first'' set of {{mvar|N}} equations <math display="inline">\mathbf{q} = -\frac{\partial G_{4}}{\partial \mathbf{p}}</math> define relations between the new generalized momenta {{math|'''P'''}} and the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. Ideally, one can invert these relations to obtain formulae for each {{math|''P<sub>k</sub>''}} as a function of the old canonical coordinates. Substitution of these formulae for the {{math|'''P'''}} coordinates into the ''second'' set of {{mvar|N}} equations <math display="inline">\mathbf{Q} = \frac{\partial G_{4}}{\partial \mathbf{P}} </math> yields analogous formulae for the new generalized coordinates {{math|'''Q'''}} in terms of the old [[canonical coordinates]] {{math|('''q''', '''p''')}}. We then invert both sets of formulae to obtain the ''old'' [[canonical coordinates]] {{math|('''q''', '''p''')}} as functions of the ''new'' [[canonical coordinates]] {{math|('''Q''', '''P''')}}. Substitution of the inverted formulae into the final equation <math display="inline">K = H + \frac{\partial G_{4}}{\partial t}</math> yields a formula for {{mvar|K}} as a function of the new [[canonical coordinates]] {{math|('''Q''', '''P''')}}.

=== Limitations on the four types of generating functions ===
Considering <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> as an example, using generating function of second kind: <math display="inline">{p}_i = \frac{\partial G_{2}}{\partial {q}_i}
 </math> and <math display="inline">{Q}_i = \frac{\partial G_{2}}{\partial {P}_i} </math>, the first set of equations consisting of variables <math display="inline">\mathbf{p} </math>, <math display="inline">\mathbf{q} </math> and <math display="inline">\mathbf{P} </math> has to be inverted to get <math display="inline">\mathbf{P}(\mathbf q, \mathbf p) </math>. This process is possible when the matrix defined by <math display="inline">a_{ij}=\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j} </math> is non-singular using the [[inverse function theorem]], and can be restated as the following relation.<ref>{{Harvnb|Johns|2005|p=438}}</ref>

<math display="block">\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|{\neq0}</math>

Hence, restrictions are placed on generating functions to have the matrices: <math display="inline">\left[\frac{\partial^2 G_1}{\partial Q_j\partial q_i} \right] </math>, <math display="inline">\left[\frac{\partial^2 G_2}{\partial P_j\partial q_i} \right]  </math>, <math display="inline">\left[\frac{\partial^2 G_3}{\partial p_j\partial Q_i} \right]  </math> and <math display="inline">\left[\frac{\partial^2 G_4}{\partial p_j\partial P_i} \right]  </math>, being non-singular.<ref>{{Harvnb|Lurie|2002|p=547}}</ref><ref>{{Harvnb|Sudarshan|Mukunda|2010|p=58}}</ref> These conditions also correspond to local invertibility of the coordinates. From these restrictions, it can be stated that type 1 and type 4 generating functions always have a non-singular <math display="inline">\left[\frac{\partial Q_i(\mathbf q,\mathbf p)}{\partial p_j} \right]  </math> matrix whereas type 2 and type 3 generating functions always have a non-singular <math display="inline">\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right]  </math> matrix. Hence, the canonical transformations resulting from these four generating functions alone are not completely general.<ref>{{Harvnb|Johns|2005|p=437-439}}</ref>

=== Generalized use of generating functions ===

In other words, since {{math|('''Q''', '''P''')}} and {{math|('''q''', '''p''')}} are each {{math|2''N''}} independent functions, it follows that to have generating function of the form <math display="inline">G_{1}(\mathbf{q}, \mathbf{Q}, t) </math> and <math>G_{4}(\mathbf{p}, \mathbf{P}, t)</math> or <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> and <math>G_{3}(\mathbf{p}, \mathbf{Q}, t)</math>, the corresponding Jacobian matrices <math display="inline">\left[\frac{\partial Q_i}{\partial p_j} \right]  </math> and <math display="inline">\left[\frac{\partial P_i}{\partial p_j} \right]  </math> are restricted to be non singular, ensuring that the generating function is a function of {{math|2''N'' + 1}} independent variables. However, as a feature of canonical transformations, it is always possible to choose {{math|2''N''}} such independent functions from sets {{math|('''q''', '''p''')}} or {{math|('''Q''', '''P''')}}, to form a generating function representation of canonical transformations, including the time variable. Hence, it can be proven that every finite canonical transformation can be given as a closed but implicit form that is a variant of the given four simple forms.<ref>{{Harvnb|Sudarshan|Mukunda|2010|pages=58-60}}</ref>

{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Proof
|-
|
Consider taking a full set of generalized coordinates <math display="inline"> \{q_{1}, q_{2}, \ldots, q_{N-1}, q_{N} \} </math> and adding to the set, while preserving local invertibility of coordinates in the set, as many transformed coordinates as possible, labelled <math display="inline">\{Q_{1}, Q_{2}, \ldots, Q_{k} \}</math> without loss of generality.

It can be shown that the set, <math display="inline">\{q_{1}, \ldots, q_{N}, Q_{1}, \ldots, Q_{k}, P_{k+1}, \ldots, P_{N} \}</math> is a set of locally independent coordinates. Proof of local invertibility of the set of coordinates is given by proving non singularity of <math display="inline">\frac{\partial(Q_1,\ldots,Q_k,P_{k+1},\ldots,P_N)}{\partial(p_1,\ldots,p_N)}</math> or the non existence of a non trivial null eigenvector such that <math display="inline">\sum_a\epsilon_a\frac{\partial Q_a}{\partial p_s}+\sum_b \eta_b\frac{\partial P_b}{\partial p_s}=0,\, \forall s</math> where the index <math display="inline">a=1,\ldots,k</math> and <math display="inline">b=k+1,\ldots,N</math>.

Letting <math display="inline"> Q_b=f_b(q_s,Q_a) </math> and assuming the existence of a null eigenvector in the following derivation:

<math display="inline"> \eta_{b'}=\sum_a\epsilon_a \{Q_{b'},Q_a\}+\sum_b\eta_b\{Q_{b'},P_b\} =\sum_s \frac{\partial f_{b'}}{\partial q_s}(\sum_a\epsilon_a\frac{\partial Q_a}{\partial p_s}+\sum_b \eta_b\frac{\partial P_b}{\partial p_s})=0 </math>

Hence all <math display="inline">\eta_b=0</math>. By condition of local invertibility it follows that for the remaining part of the equation, <math display="inline"> \sum \frac{\partial Q_a}{\partial p_i}\epsilon_i= \delta Q_a(p_1,\ldots,p_N) =0\implies \epsilon_i =0 \quad \forall\, a=1,\ldots,k </math> thereby showing that the only null eigenvector <math display="inline">\frac{\partial(Q_1,\ldots,Q_k,P_{k+1},\ldots,P_N)}{\partial(p_1,\ldots,p_N)}</math> is the trivial vector implying that it is a non singular matrix. Hence it is shown that it is possible to take sets such as <math display="inline">\{q_{1}, \ldots, q_{N}, Q_{1}, \ldots, Q_{k},P_{k+1}, \ldots, P_{N} \}</math> that is a combination of new and old coordinates that preserves the {{math|2''N''}} independent variables property which can be used to interpret any coordinate transform as arising from a generating function on these set of coordinates.
|}