Weyl's lemma (Laplace equation): Difference between revisions

Revision as of 10:32, 12 January 2014

The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form

\sum_{i} p_{i} δ q_{i} = \sum_{i} P_{i} δ 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} Ω_{1} (q_{1}, q_{2}, \dots, q_{n}, Q_{1}, Q_{2}, \dots Q_{n}) = 0 \\ \dots \\ Ω_{m} (q_{1}, q_{2}, \dots, q_{n}, Q_{1}, Q_{2}, \dots Q_{n}) = 0 \end{aligned}

where $1 < m \leq n$ . When $m = n$ a Mathieu transformation becomes a Lagrange point transformation.

References

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Template:Classicalmechanics-stub

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+The '''Mathieu transformations''' make up a subgroup of [[canonical transformation]]s preserving the [[differential form]]
+:<math>\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 [[Invariant (mathematics)|invariance]], there should exist at least one [[relation (mathematics)|relation]] between <math>q_i</math> and <math>Q_i</math> '''only''' (without any <math>p_i,P_i</math> involved).
+:<math>
+\begin{align}
+\Omega_1(q_1,q_2,\ldots,q_n,Q_1,Q_2,\ldots Q_n)=0\\
+\ldots\\
+\Omega_m(q_1,q_2,\ldots,q_n,Q_1,Q_2,\ldots Q_n)=0
+\end{align}
+</math>
+where <math>1 < m \le n</math>. When <math>m=n</math> a Mathieu transformation becomes a [[Lagrange point transformation]].
+== See also ==
+* [[Canonical transformation]]
+== References ==
+* {{cite book | author=[[Lanczos]], Cornelius | title=The Variational Principles of Mechanics | location= Toronto | publisher=University of Toronto Press | year=1970 | isbn=0-8020-1743-6}}
+* {{cite book | author=[[Edmund Whittaker|Whittaker]], Edmund | title=A Treatise on the Analytical Dynamics of Particles and Rigid Bodies }}
+[[Category:Mechanics]]
+[[Category:Hamiltonian mechanics]]
+{{classicalmechanics-stub}}

Weyl's lemma (Laplace equation): Difference between revisions

Revision as of 10:32, 12 January 2014

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