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| The '''[[Constantin Carathéodory|Carathéodory]]–[[Carl Gustav Jakob Jacobi|Jacobi]]–[[Sophus Lie|Lie]] theorem''' is a [[theorem]] in [[symplectic geometry]] which generalizes [[Darboux's theorem]].
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| ==Statement==
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| 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
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| :<math>df_1(p) \wedge \ldots \wedge df_r(p) \neq 0,</math>
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| 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 | |
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| :<math>\omega = \sum_{i=1}^n df_i \wedge dg_i.</math> | |
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| ==Applications==
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| 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''.
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| ==References==
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| * Lee, John M., ''Introduction to Smooth Manifolds'', Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.
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| {{DEFAULTSORT:Caratheodory-Jacobi-Lie theorem}}
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| [[Category:Symplectic geometry]]
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| [[Category:Theorems in differential geometry]]
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| {{differential-geometry-stub}}
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