Physics:Mathieu transformation
From HandWiki
The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form
- [math]\displaystyle{ \sum_i p_i \delta q_i=\sum_i P_i \delta Q_i \, }[/math]
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 [math]\displaystyle{ q_i }[/math] and [math]\displaystyle{ Q_i }[/math] only (without any [math]\displaystyle{ p_i,P_i }[/math] involved).
- [math]\displaystyle{ \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} }[/math]
where [math]\displaystyle{ 1 \lt m \le n }[/math]. When [math]\displaystyle{ m=n }[/math] a Mathieu transformation becomes a Lagrange point transformation.
See also
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.
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