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).

${\begin{aligned}\Omega _{1}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&=0\\&{}\ \ \vdots \\\Omega _{m}(q_{1},q_{2},\ldots ,q_{n},Q_{1},Q_{2},\ldots Q_{n})&=0\end{aligned}}$

where $1<m\leq n$. 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.