Carathéodory-Jacobi-Lie theorem

The Carathéodory-Jacobi-Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.

tatement

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

:$df\_1(p)\; wedge\; ldots\; wedge\; df\_r(p)\; eq\; 0,$

where {f_i, f_j} = 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

:$omega\; =\; sum\_\{i=1\}^\{n\}\; df\_i\; wedge\; dg\_i$.

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 $dH\; eq\; 0$ there is a 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.