Editing Canonical transformation (section)

=== 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>