Editing Canonical transformation (section)

==== 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.