Editing Canonical transformation (section)

=== Canonical transformation relations ===
By solving for: 

<math display="block">\lambda \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) \right] = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{dG}{dt} </math> 

with various forms of generating function, the relation between K and H goes as <math display="inline">\frac{\partial G}{\partial t} = K-\lambda H </math> instead, which also applies for <math display="inline">\lambda = 1 </math> case. 

All results presented below can also be obtained by replacing <math display="inline">q \rightarrow \sqrt{\lambda}q </math>, <math display="inline">p \rightarrow \sqrt{\lambda}p </math> and <math display="inline">H \rightarrow {\lambda}H </math> from known solutions, since it retains the form of [[Hamilton's equations]]. The extended canonical transformations are hence said to be result of a canonical transformation (<math display="inline">\lambda = 1 </math>) and a trivial canonical transformation (<math display="inline">\lambda \neq 1 </math>) which has <math display="inline">M J M^T = \lambda J
</math> (for the given example, <math display="inline">M = \sqrt{\lambda} I
</math> which satisfies the condition).<ref>{{Harvnb|Giacaglia|1972|p=18-19}}</ref>

Using same steps previously used in previous generalization, with <math display="inline">\frac{\partial G}{\partial t} = K-\lambda H </math> in the general case, and retaining the equation <math display="inline">J \left(\nabla_\varepsilon \frac{\partial g}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t}  </math>, extended canonical transformation partial differential relations are obtained as:

<math display="block">\begin{align}
\left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= -\lambda \left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\
\left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \lambda \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} 
\end{align}</math>

<math display="block">\begin{align}
\left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \lambda \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\
\left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= -\lambda  \left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t}
\end{align}</math>