Editing Canonical transformation (section)

=== Invariance of the Lagrange bracket ===
The [[Lagrange bracket]] which is defined as:

<math display="block">
[ u, v ]_{\eta}  := \sum_{i=1}^n \left(\frac{\partial q_i}{\partial u} \frac{\partial p_i}{\partial v} - \frac{\partial p_i}{\partial u} \frac{\partial q_i}{\partial v }  \right)
</math>

can be represented in matrix form as:

<math display="block">
[ u, v ]_{\eta}  := \left(\frac {\partial \eta}{\partial u}\right)^T J \left(\frac {\partial \eta}{\partial v}\right) 
</math>

Using similar derivation, gives:

<math display="block">[u, v]_\varepsilon = (\partial_u  \varepsilon )^T \,J\, (\partial_v \varepsilon) = (M \, \partial_u  \eta )^T \,J \,
( M \,\partial_v \eta) = (\partial_u  \eta )^T\, M^TJ M\,
(\partial_v \eta) = (\partial_u  \eta )^T\, J\,(\partial_v \eta) = [u, v]_\eta</math>

The symplectic condition can also be recovered by taking <math display="inline">u=\eta_i
</math> and <math display="inline">v=\eta_j 
</math> which shows that <math display="inline">(M^T J M )_{ij}= J_{i j} 
</math>. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that <math display="inline">\mathcal L_{ij}(\eta) =[\eta_i,\eta_j]_\varepsilon=(M^T J M )_{ij} 
</math>, which is also the result of explicitly calculating the matrix element by expanding it.<ref name=":0" />