Let $M$ be a symplectic manifold with symplectic form $\Omega$. The Poisson bracket is a bilinear operation on the set of differentiable functions on $M$. In terms of local Darboux coordinates $p_{1},\ldots,p_{n},q_{1},\ldots,q_{n}$, the Poisson bracket of two functions is defined as follows:

It can be shown that the value of $[f,g]$ does not depend on the choice of Darboux coordinates. Therefore, the Poisson bracket is a well-defined operation on the symplectic manifold. Also, some authors use a different sign convention — what they call $[f,g]$ is what would be referred to as $-[f,g]$ here.

The Poisson bracket can be defined without reference to a special coordinate system as follows:

Here $\Omega^{{-1}}$ is the inverse of the symplectic form, and its components in an arbitrary coordinate system are denoted $\Omega^{{ij}}$.

The Poisson bracket sastisfies several important algebraic identities. It is antisymmetric:

It is a derivation:

It satisfies Jacobi’s identitity:

The Hamilton equations can be expressed elegantly in terms of the Poisson bracket. If $X$ is a smooth function on $M$, we can describe the time-evolution of $X$ by the equation

If $X$ is a smooth function on $\mathbb{R}\times M$, we can describe the time-evolution of $X$ by the more general equation