Poisson bracket

Let $M$ be a symplectic manifold with symplectic form $\mathrm{\Omega }$. The Poisson bracket is a bilinear operation on the set of differentiable functions on $M$. In terms of local Darboux coordinates $p_1,\mathrm{\dots },p_n,q_1,\mathrm{\dots },q_n$, the Poisson bracket of two functions is defined as follows:

$$[f,g]=\sum _{i=1}^n\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}$$

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:

$$[f,g]={\mathrm{\Omega }}^{-1}(df,dg)=\sum _{i=1}^{2n}{\mathrm{\Omega }}^{ij}\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial x_j}$$

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

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

$$[f,g]=-[g,f]$$

It is a derivation:

$$[fg,h]=f[g,h]+g[f,h]$$

It satisfies Jacobi’s identitity:

$$[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0$$

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

$$\frac{dX}{dt}=[X,H]$$

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

$$\frac{dX}{dt}=\frac{\partial X}{\partial t}-[X,H]$$