Editing Canonical transformation (section)

=== Explicit construction ===
Consider the following generating function:

<math display="block">G_2(q,P,t)= qP + \alpha G(q,P,t)  </math>

Since for <math>\alpha=0 </math>, <math>G_2 = qP </math> has the resulting canonical transformation, <math>Q = q </math> and <math>P = p </math>, this type of generating function can be used for infinitesimal canonical transformation by restricting <math>\alpha </math> to an infinitesimal value. 

From the conditions of generators of second type:

<math display="block">\begin{align}
{p} &= \frac{\partial G_{2}}{\partial {q}} = P + \alpha  \frac{\partial G}{\partial {q}} (q,P,t) \\
{Q} &= \frac{\partial G_{2}}{\partial {P}} = q + \alpha  \frac{\partial G}{\partial {P}}  (q,P,t) \\
\end{align}</math>


Since <math>P = P(q,p,t;\alpha) </math>, changing the variables of the function <math>G </math> to <math>G(q,p,t) </math> and neglecting terms of higher order of <math>\alpha </math>, gives:<ref>{{Harvnb|Johns|2005|p=452-454}}</ref>

<math display="block">\begin{align}
{p} &= P + \alpha  \frac{\partial G}{\partial {q}} (q,p,t) \\
{Q} &= q + \alpha  \frac{\partial G}{\partial p}  (q,p,t) \\
\end{align}</math>

Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition.<ref name=":1">{{Cite web |last=Hergert |first=Heiko |date=December 10, 2021 |title=PHY422/820: Classical Mechanics |url=https://people.nscl.msu.edu/~hergert/phy820/material/pdfs/w14.pdf |url-status=live |archive-url=https://web.archive.org/web/20231222161338/https://people.nscl.msu.edu/~hergert/phy820/material/pdfs/w14.pdf |archive-date=December 22, 2023 |access-date=December 22, 2023}}</ref> The function <math>G(q,p,t) </math> is very significant in infinitesimal canonical transformations and is referred to as the generator of infinitesimal canonical transformation.