The Mathieu transformations make up a subgroup of

canonical transformation In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canoni ...

s preserving the

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

\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 Émile Léonard Mathieu (; 15 May 1835, in Metz – 19 October 1890, in Nancy) was a French mathematician. He is known for his work in group theory and mathematical physics. He has given his name to the Mathieu functions, Mathieu groups and Mathi ...

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In order to have this invariance, there should exist at least one

relation Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...

between

q_i

and

Q_i

only (without any

p_i,P_i

involved). :

\begin
\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

where

1 < m \le n

. When

m=n

a Mathieu transformation becomes a

Lagrange point transformation In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canon ...

References

* * Mechanics Hamiltonian mechanics {{classicalmechanics-stub

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See also

References