Mathieu transformation explained

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.

See also

• Canonical transformation

References

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