Editing Canonical transformation (section)

==== Active view of transformation ====
Thus, using the relations from infinitesimal canonical transformations, the change in the system states under active view of the canonical transformation is said to be:

<math display="block">\begin{align}
& \delta q = \alpha \frac{\partial G}{\partial p}  (q,p,t)   \quad \text{and} \quad \delta p = - \alpha  \frac{\partial G}{\partial q}  (q,p,t) , \\  
\end{align}  </math>


or as <math>\delta \eta = \alpha J \nabla_\eta G </math> in matrix form.


For any function <math>u(\eta)   </math>, it changes under active view of the transformation according to:

<math display="block">\delta u = u(\eta +\delta \eta)-u(\eta) = (\nabla_\eta u)^T\delta\eta=\alpha (\nabla_\eta u)^T J (\nabla_\eta G) = \alpha \{ u,G \}   . </math>