\{z^{\alpha}, H \}$ where $\{
\cdot , \cdot \} \colon C^{\infty} ({\rm M}) \times C^{\infty} ({\rm M})
\rightarrow C^{\infty} ({\rm M}).$

The bracket that appears in the equation of motion is a Poisson bracket if it is antisymmetric, bilinear, and it satisfies two more condtions. These are the Leibnitz rule, $\{ f,gh \} = g \{ f,h \} + \{ f,g \} h$ and the Jacobi identity,

$$\label{JACOBI}
\{ f, \{ g,h \} \} + \{ g, \{ h,f \} \} + \{ h, \{ f,g \} \} = 0.$$

The most familiar example of a bracket satisfying these properties is given by the so-called canonical bracket:

$$\label{CANBRAC}
\{ f, g \} = \sum_{i=1}^n {\frac{\partial f }{\partial q^i}} {\frac{\partial
g }{\partial p_i}} -{\frac{\partial g }{\partial q^i}} {\frac{\partial f }{%
\partial p_i}}.$$

But there are the other, noncanonical brackets that obey the above conditions; some are given in the examples. However, such noncanonical brackets, even when they are degenerate, can be locally represented in the form (\[CANBRAC\]) (where $n$ is equal to half the rank of the bracket), as shown by the Darboux theorem (see also [@wei1]). In this sense, they are true generalizations of the more familiar canonical brackets.

By 