Physics: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).

$\begin{array}{rl}{\mathrm{\Omega }}_1(q_1,q_2,\dots ,q_n,Q_1,Q_2,\dots Q_n)& =0\\ & ⋮\\ {\mathrm{\Omega }}_m(q_1,q_2,\dots ,q_n,Q_1,Q_2,\dots Q_n)& =0\end{array}$

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.