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\\ \ldots\\ \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.

See also

Canonical transformation

References

Lanczos, Cornelius (1970). The Variational Principles of Mechanics. Toronto: University of Toronto Press. ISBN 0-8020-1743-6.
Whittaker, Edmund. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies.