Editing Canonical transformation (section)

=== Poisson and Lagrange brackets ===
The Poisson brackets are changed as follows:

<math display="block">\{u, v\}_\eta = (\nabla_\eta u)^T J (\nabla_\eta v) = (M^T \nabla_\varepsilon u)^T J (M^T \nabla_\varepsilon v) = (\nabla_\varepsilon u)^T M J M^T (\nabla_\varepsilon v) =  \lambda (\nabla_\varepsilon u)^T J (\nabla_\varepsilon v) = \lambda \{u, v\}_\varepsilon</math>

whereas, the Lagrange brackets are changed as:

<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) = \lambda  (\partial_u  \eta )^T\, J\,(\partial_v \eta) = \lambda [u, v]_\eta</math>

Hence, the Poisson bracket scales by the inverse of <math display="inline">\lambda
</math> whereas the Lagrange bracket scales by a factor of <math display="inline">\lambda
</math>.<ref>{{Harvnb|Giacaglia|1972|p=16}}-17</ref>