Editing Canonical transformation (section)

=== Indirect conditions ===
Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate {{math|''Q<sub>m</sub>''}} is

<math display="block">\begin{align}
\dot{Q}_{m} &= \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} \\
&= \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \frac{\partial H}{\partial \mathbf{p}} - \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \frac{\partial H}{\partial \mathbf{q}} \\
&= \lbrace Q_m , H \rbrace
\end{align}</math><br />
where {{math|{⋅, ⋅} }} is the [[Poisson bracket]].


Similarly for the identity for the conjugate momentum, ''P<sub>m</sub>'' using the form of the "Kamiltonian" it follows that:

<math display="block">\begin{align}
\frac{\partial K(\mathbf{Q}, \mathbf{P}, t)}{\partial P_{m}}
&= \frac{\partial K(\mathbf{Q}(\mathbf{q}, \mathbf{p}), \mathbf{P}(\mathbf{q}, \mathbf{p}), t)}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}}
 + \frac{\partial K(\mathbf{Q}(\mathbf{q}, \mathbf{p}), \mathbf{P}(\mathbf{q}, \mathbf{p}), t)}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \\[1ex]
&= \frac{\partial H(\mathbf{q}, \mathbf{p}, t)}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}}
 + \frac{\partial H(\mathbf{q}, \mathbf{p}, t)}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \\[1ex]
&= \frac{\partial H}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}}
 + \frac{\partial H}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}}
\end{align}</math>


Due to the form of the Hamiltonian equations of motion,

<math display="block">\begin{align}
\dot{\mathbf{P}} &= -\frac{\partial K}{\partial \mathbf{Q}}\\
\dot{\mathbf{Q}} &= \,\,\,\, \frac{\partial K}{\partial \mathbf{P}} 
\end{align}</math>

if the transformation is canonical, the two derived results must be equal, resulting in the equations:

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

The analogous argument for the generalized momenta ''P<sub>m</sub>'' leads to two other sets of equations:

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

These are the '''indirect conditions''' to check whether a given transformation is canonical.