Polyconic projection: Difference between revisions
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The | The '''[[Constantin Carathéodory|Carathéodory]]–[[Carl Gustav Jakob Jacobi|Jacobi]]–[[Sophus Lie|Lie]] theorem''' is a [[theorem]] in [[symplectic geometry]] which generalizes [[Darboux's theorem]]. | ||
==Statement== | |||
Let ''M'' be a 2''n''-dimensional [[symplectic manifold]] with symplectic form ω. For ''p'' ∈ ''M'' and ''r'' ≤ ''n'', let ''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>r</sub> be [[smooth function]]s defined on an [[open neighborhood]] ''V'' of ''p'' whose [[differential form|differential]]s are [[linearly independent]] at each point, or equivalently | |||
:<math>df_1(p) \wedge \ldots \wedge df_r(p) \neq 0,</math> | |||
where {f<sub>i</sub>, f<sub>j</sub>} = 0. (In other words they are pairwise in involution.) Here {–,–} is the [[Poisson bracket]]. Then there are functions ''f''<sub>r+1</sub>, ..., ''f''<sub>n</sub>, ''g''<sub>1</sub>, ''g''<sub>2</sub>, ..., ''g''<sub>n</sub> defined on an open neighborhood ''U'' ⊂ ''V'' of ''p'' such that (f<sub>i</sub>, g<sub>i</sub>) is a [[symplectic chart]] of ''M'', i.e., ω is expressed on ''U'' as | |||
:<math>\omega = \sum_{i=1}^n df_i \wedge dg_i.</math> | |||
==Applications== | |||
As a direct application we have the following. Given a [[Hamiltonian system]] as <math>(M,\omega,H)</math> where ''M'' is a symplectic manifold with symplectic form <math>\omega</math> and ''H'' is the [[Hamiltonian mechanics|Hamiltonian function]], around every point where <math>dH \neq 0</math> there is a symplectic chart such that one of its coordinates is ''H''. | |||
==References== | |||
* Lee, John M., ''Introduction to Smooth Manifolds'', Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds. | |||
{{DEFAULTSORT:Caratheodory-Jacobi-Lie theorem}} | |||
[[Category:Symplectic geometry]] | |||
[[Category:Theorems in differential geometry]] | |||
{{differential-geometry-stub}} | |||
Revision as of 18:36, 22 December 2013
The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.
Statement
Let M be a 2n-dimensional symplectic manifold with symplectic form ω. For p ∈ M and r ≤ n, let f1, f2, ..., fr be smooth functions defined on an open neighborhood V of p whose differentials are linearly independent at each point, or equivalently
where {fi, fj} = 0. (In other words they are pairwise in involution.) Here {–,–} is the Poisson bracket. Then there are functions fr+1, ..., fn, g1, g2, ..., gn defined on an open neighborhood U ⊂ V of p such that (fi, gi) is a symplectic chart of M, i.e., ω is expressed on U as
Applications
As a direct application we have the following. Given a Hamiltonian system as where M is a symplectic manifold with symplectic form and H is the Hamiltonian function, around every point where there is a symplectic chart such that one of its coordinates is H.
References
- Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.