Editing Canonical transformation (section)

=== Active and passive transformations ===
{{See also|Active and passive transformation}}
In the active view of transformations, the coordinate system is changed without the physical system changing, whereas in the passive view of transformation, the coordinate system is retained and the physical system is said to undergo transformations.

==== 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>

==== Passive view of transformation ====
Considering the change of Hamiltonians in the [[Active and passive transformation|passive view]], i.e., for a fixed point,<math display="block">K(Q=q_0,P=p_0,t) - H(q=q_0,p=p_0,t) = \left(H(q_0',p_0',t) + \frac{\partial G_{2}}{\partial t}\right) - H(q_0,p_0,t) = - \delta H +\alpha \frac{\partial G}{\partial t} = \alpha\left(\{ G,H\}+\frac{\partial G}{\partial t} \right)=\alpha\frac{dG}{dt}    </math>

where <math display="inline">(q=q_0',p=p_0') </math> are mapped to the point, <math display="inline">(Q=q_0,P=p_0) </math> by the infinitesimal canonical transformation, and similar change of variables for <math>G(q,P,t) </math> to <math>G(q,p,t) </math> is considered up-to first order of <math>\alpha </math>. Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.