Mathieu transformation

The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form


 * $$\sum_i p_i \delta q_i=\sum_i P_i \delta Q_i \,$$

The transformation is named after the French mathematician Émile Léonard Mathieu.

Details
In order to have this invariance, there should exist at least one relation between $$q_i$$ and $$Q_i$$ only (without any $$p_i,P_i$$ involved).



\begin{align} \Omega_1(q_1,q_2,\ldots,q_n,Q_1,Q_2,\ldots Q_n) & =0 \\ & {}\ \  \vdots\\ \Omega_m(q_1,q_2,\ldots,q_n,Q_1,Q_2,\ldots Q_n) & =0 \end{align} $$

where $$1 < m \le n$$. When $$m=n$$ a Mathieu transformation becomes a Lagrange point transformation.