Editing Canonical transformation (section)

== Conditions for restricted canonical transformation ==
Restricted canonical transformations are coordinate transformations where transformed coordinates {{math|'''Q'''}} and {{math|'''P'''}} do not have explicit time dependence, i.e., <math display="inline">\mathbf Q=\mathbf Q(\mathbf q,\mathbf  p)</math> and <math display="inline">\mathbf  P=\mathbf P(\mathbf q,\mathbf p) </math>. The functional form of [[Hamilton's equations]] is

<math display="block">\begin{align}
\dot{\mathbf{p}} &= -\frac{\partial H}{\partial \mathbf{q}} \,,
&
\dot{\mathbf{q}} &= \frac{\partial H}{\partial \mathbf{p}}
\end{align}</math>

In general, a transformation {{math|('''q''', '''p''') → ('''Q''', '''P''')}} does not preserve the form of [[Hamilton's equations]] but in the absence of time dependence in transformation, some simplifications are possible. Following the formal definition for a canonical transformation, it can be shown that for this type of transformation, the new Hamiltonian (sometimes called the Kamiltonian<ref>{{harvnb|Goldstein|Poole|Safko|2007|p=370}}</ref>)  can be expressed as:

<math display="block">K(\mathbf Q, \mathbf P, t)= H(q(\mathbf Q,\mathbf P),p(\mathbf Q,\mathbf P),t) + \frac{\partial G}{\partial t}(t)</math>

where it differs by a partial time derivative of a function known as a generator, which reduces to being only a function of time for restricted canonical transformations. 

In addition to leaving the form of the Hamiltonian unchanged, it is also permits the use of the unchanged Hamiltonian in the Hamilton's equations of motion due to the above form as:

<math display="block">\begin{alignat}{3}
\dot{\mathbf{P}} &= -\frac{\partial K}{\partial \mathbf{Q}} &&= -\left(\frac{\partial H}{\partial \mathbf{Q}}\right)_{\mathbf Q,\mathbf P,t}\\
\dot{\mathbf{Q}} &= \,\,\,\, \frac{\partial K}{\partial \mathbf{P}} &&= \,\,\,\, \, \left(\frac{\partial H}{\partial \mathbf{P}}\right)_{\mathbf Q,\mathbf P ,t}\\
\end{alignat}</math>

Although canonical transformations refers to a more general set of transformations of phase space corresponding with less permissive transformations of the Hamiltonian, it provides simpler conditions to obtain results that can be further generalized. All of the following conditions, with the exception of bilinear invariance condition, can be generalized for canonical transformations, including time dependance.

=== Indirect conditions ===
Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate {{math|''Q<sub>m</sub>''}} is

<math display="block">\begin{align}
\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{align}</math><br />
where {{math|{⋅, ⋅} }} is the [[Poisson bracket]].


Similarly for the identity for the conjugate momentum, ''P<sub>m</sub>'' using the form of the "Kamiltonian" it follows that:

<math display="block">\begin{align}
\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}} \\[1ex]
&= \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}} \\[1ex]
&= \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{align}</math>


Due to the form of the Hamiltonian equations of motion,

<math display="block">\begin{align}
\dot{\mathbf{P}} &= -\frac{\partial K}{\partial \mathbf{Q}}\\
\dot{\mathbf{Q}} &= \,\,\,\, \frac{\partial K}{\partial \mathbf{P}} 
\end{align}</math>

if the transformation is canonical, the two derived results must be equal, resulting in the equations:

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

The analogous argument for the generalized momenta ''P<sub>m</sub>'' leads to two other sets of equations:

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

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

=== Symplectic condition ===
Sometimes the Hamiltonian relations are represented as:

<math display="block">\dot{\eta}= J \nabla_\eta H
</math>

Where <math display="inline">J := \begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix},</math> 

and <math display="inline">\mathbf{\eta} =
  \begin{bmatrix}
    q_1\\
    \vdots \\
    q_n\\
    p_1\\
    \vdots\\
    p_n\\    
  \end{bmatrix}
</math>. Similarly, let <math display="inline">\mathbf{\varepsilon} =
  \begin{bmatrix}
    Q_1\\
    \vdots \\
    Q_n\\
    P_1\\
    \vdots\\
    P_n\\    
  \end{bmatrix}
</math>.



From the relation of partial derivatives, converting the <math>\dot{\eta}= J \nabla_\eta H
</math> relation in terms of partial derivatives with new variables gives <math>\dot{\eta}=J ( M^T \nabla_\varepsilon H)
</math> where <math display="inline">M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}</math>. Similarly for <math display="inline">\dot{\varepsilon}
</math>,

<math display="block">\dot{\varepsilon}=M\dot{\eta} =M J M^T \nabla_\varepsilon H
</math>


Due to form of the Hamiltonian equations for <math display="inline">\dot{\varepsilon}
</math>,

<math display="block">\dot{\varepsilon}=J \nabla_\varepsilon K  =  J \nabla_\varepsilon H
</math>


where <math display="inline">\nabla_\varepsilon K  =  \nabla_\varepsilon H
</math> can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:<ref> {{Harvnb|Goldstein|Poole|Safko|2007|p=381-384}}</ref>

<math display="block">M J M^T = J
</math>

The left hand side of the above is called the Poisson matrix of <math>\varepsilon
</math>, denoted as <math display="inline">\mathcal P(\varepsilon) = MJM^T
</math>. Similarly, a Lagrange matrix of <math>\eta
</math> can be constructed as <math display="inline">\mathcal L(\eta) = M^TJM   
</math>.<ref name=":0">{{Harvnb|Giacaglia|1972|p=8-9}}</ref> It can be shown that the symplectic condition is also equivalent to <math display="inline">M^T J M = J  
</math> by using the <math display="inline">J^{-1}=-J
</math> property. The set of all matrices <math display="inline">M
</math> which satisfy symplectic conditions form a [[symplectic group]]. The symplectic conditions are equivalent with indirect conditions as they both lead to the equation <math display="inline">\dot{\varepsilon}=  J \nabla_\varepsilon H
</math>, which is used in both of the derivations.

=== Invariance of the Poisson bracket ===
The [[Poisson bracket]] which is defined as:<math display="block">\{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)</math>can be represented in matrix form as:

<math display="block">\{u, v\}_\eta := (\nabla_\eta u)^T J (\nabla_\eta v)</math>

Hence using partial derivative relations and symplectic condition gives:<ref>{{Harvnb|Lemos|2018|p=255}}</ref><math display="block">\{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 M J M^T (\nabla_\varepsilon v) = (\nabla_\varepsilon u)^T J (\nabla_\varepsilon v) = \{u, v\}_\varepsilon</math>

The symplectic condition can also be recovered by taking <math display="inline">u=\varepsilon_i
</math> and <math display="inline">v=\varepsilon_j
</math> which shows that <math display="inline">(M J M^T )_{ij}= J_{i j}
</math>. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that <math display="inline">\mathcal P_{ij}(\varepsilon) = \{ \varepsilon_i,\varepsilon_j\}_\eta =(M J M^T )_{ij}
</math>, which is also the result of explicitly calculating the matrix element by expanding it.<ref name=":0" />    

=== Invariance of the Lagrange bracket ===
The [[Lagrange bracket]] which is defined as:

<math display="block">
[ 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)
</math>

can be represented in matrix form as:

<math display="block">
[ u, v ]_{\eta}  := \left(\frac {\partial \eta}{\partial u}\right)^T J \left(\frac {\partial \eta}{\partial v}\right) 
</math>

Using similar derivation, gives:

<math display="block">[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^TJ M\,
(\partial_v \eta) = (\partial_u  \eta )^T\, J\,(\partial_v \eta) = [u, v]_\eta</math>

The symplectic condition can also be recovered by taking <math display="inline">u=\eta_i
</math> and <math display="inline">v=\eta_j 
</math> which shows that <math display="inline">(M^T J M )_{ij}= J_{i j} 
</math>. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that <math display="inline">\mathcal L_{ij}(\eta) =[\eta_i,\eta_j]_\varepsilon=(M^T J M )_{ij} 
</math>, which is also the result of explicitly calculating the matrix element by expanding it.<ref name=":0" />

=== 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:<ref>{{Harvnb|Hand|Finch|1999|p=250-251}}</ref>

<math display="inline"> d \varepsilon=( dq_1, dp_{1},0,0,\ldots),\quad\delta \varepsilon=(\delta q_{1},\delta p_{1},0,0,\ldots).    </math>


The area of the infinitesimal parallelogram is given by:

<math display="inline"> \delta a(12)=d q_{1}\delta p_{1}-\delta q_{1} d p_{1}={(\delta\varepsilon)}^T\,J \, d \varepsilon.  </math>


It follows from the <math display="inline">M^T J M = J  
</math> symplectic condition that the infinitesimal area is conserved under canonical transformation:

<math display="inline"> \delta a(12)={(\delta\varepsilon)}^T\,J \,d \varepsilon={(M\delta\eta)}^T\,J \,Md \eta= {(\delta\eta)}^T\,M^TJM \,d \eta =  {(\delta\eta)}^T\,J \,d\eta = \delta A(12). </math>

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 <math display="inline"> {(d\varepsilon)}^T\,J \, \delta \varepsilon </math> under canonical transformation, expanded as:

<math display="block"> \sum \delta q \cdot dp - \delta p \cdot dq = \sum \delta Q \cdot dP - \delta P \cdot dQ </math>

If the above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met.<ref>{{harvnb|Lanczos|2012|p=121}}</ref><ref>{{harvnb|Gupta|Gupta|2008|p=304}}</ref> 
The form of the equation, <math display="inline"> {v}^T\,J \, w </math> is also known as a symplectic product of the vectors <math display="inline"> {v} </math> and <math display="inline"> w </math> and the bilinear invariance condition can be stated as a local conservation of the symplectic product.<ref>{{Harvnb|Lurie|2002|p=337}}</ref>