The Carathéodory

Constantin Carathéodory

Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...

–Jacobi

Carl Gustav Jakob Jacobi

Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...

–Lie

Sophus Lie

Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...

 theorem is a theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

 in symplectic geometry which generalizes Darboux's theorem

Darboux's theorem

Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry...

.

Statement

Let M be a 2n-dimensional symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

 with symplectic form ω. For p ∈ M and r ≤ n, let f₁, f₂, ..., f_r be smooth function

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

s defined on an open neighborhood V of p whose differential

Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s are linearly independent at each point, or equivalently


where {f_i, f_j} = 0. (In other words they are pairwise in involution.) Here {–,–} is the Poisson bracket

Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...

. 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

Applications

As a direct application we have the following. Given a Hamiltonian system

Hamiltonian system

In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....

 as where M is a symplectic manifold with symplectic form and H is the Hamiltonian function

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

, around every point where there is a chart such that one of its coordinates is H.