- Source: Mathieu transformation
The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form
∑
i
p
i
δ
q
i
=
∑
i
P
i
δ
Q
i
{\displaystyle \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
{\displaystyle q_{i}}
and
Q
i
{\displaystyle Q_{i}}
only (without any
p
i
,
P
i
{\displaystyle p_{i},P_{i}}
involved).
Ω
1
(
q
1
,
q
2
,
…
,
q
n
,
Q
1
,
Q
2
,
…
Q
n
)
=
0
⋮
Ω
m
(
q
1
,
q
2
,
…
,
q
n
,
Q
1
,
Q
2
,
…
Q
n
)
=
0
{\displaystyle {\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
≤
n
{\displaystyle 1
. When
m
=
n
{\displaystyle 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.
Kata Kunci Pencarian:
- Karim Benzema
- Mathieu transformation
- Émile Léonard Mathieu
- Mathieu
- Canonical transformation
- Mathieu function
- Index of physics articles (M)
- Semilinear map
- Almost Mathieu operator
- Group action
- Projective linear group
