Carathéodory–Jacobi–Lie theorem

Short description: Theorem in symplectic geometry which generalizes Darboux's theorem

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, ..., f_r be smooth functions defined on an open neighborhood V of p whose differentials are linearly independent at each point, or equivalently

[math]\displaystyle{ df_1(p) \wedge \ldots \wedge df_r(p) \neq 0, }[/math]

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, g₁, g₂, ..., 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

[math]\displaystyle{ \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]\displaystyle{ (M,\omega,H) }[/math] where M is a symplectic manifold with symplectic form [math]\displaystyle{ \omega }[/math] and H is the Hamiltonian function, around every point where [math]\displaystyle{ dH \neq 0 }[/math] there is a symplectic chart such that one of its coordinates is H.

References

• Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9981-8. 
• Libermann, P.; Marle, Charles-Michel (6 December 2012). Symplectic Geometry and Analytical Mechanics. ISBN 9789400938076. https://books.google.com/books?id=bWnoCAAAQBAJ&dq&pg=PA51.