Carathéodory–Jacobi–Lie Theorem

The Carathéodory– Jacobi– 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''₁, ''f''₂, ..., ''f''_r be smooth functions defined on an open neighborhood ''V'' of ''p'' whose differentials are linearly independent at each point, or equivalently

:$df_1(p) \wedge \ldots \wedge df_r(p) \neq 0,$

where = 0. (In other words, they are pairwise in involution.) Here is the Poisson bracket. Then there are functions ''f''_r+1, ..., ''f''_n, ''g''₁, ''g''₂, ..., ''g''_n defined on an open neighborhood ''U'' ⊂ ''V'' of ''p'' such that (f_i, g_i) is a symplectic chart of ''M'', i.e., ω is expressed on ''U'' as

:$\omega = \sum_^n df_i \wedge dg_i.$

Applications

As a direct application we have the following. Given a Hamiltonian system as $(M,\omega,H)$ where ''M'' is a symplectic manifold with symplectic form $\omega$ and ''H'' is the Hamiltonian function, around every point where $dH \neq 0$ 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) . Graduate-level textbook on smooth manifolds.

Symplectic geometry
Theorems in differential geometry

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