Carathéodory–Jacobi–Lie theorem explained

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

df _{1(p)} \wedge \ldots \wedge df _{r(p)} ≠ 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, 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

\omega = \sum ^{n} _{i =1} 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 ≠ 0 there is a symplectic chart such that one of its coordinates is H.

References

• Book: 10.1007/978-1-4419-9982-5 . Introduction to Smooth Manifolds . Graduate Texts in Mathematics . 2012 . 218 . 978-1-4419-9981-8. John M.. Lee .
• Book: [{{Google books|bWnoCAAAQBAJ&dq|page=51|plainurl=yes}} Symplectic Geometry and Analytical Mechanics ]. 9789400938076 . Libermann . P. . Marle . Charles-Michel . 6 December 2012 .