virtue of the above properties, a Poisson bracket defines a second rank tensor called the cosymplectic form, ${\rm J}\colon {\rm T^*M} \rightarrow 
{\rm TM}$, in the following way

$$\label{JDEF}
\langle {\rm d}f(z), {\rm J}(z) {\rm d}g(z) \rangle \colon = \{ f, g \} (z)$$

where the angular brackets denote the pairing between the cotangent and tangent spaces at the point $z \in {\rm M}.$ In a given set of coordinates $%
z^{\alpha },$ for a finite-dimensional system, equation (\[JDEF\]) has the form

$$\label{JDEF2}
{\frac{\partial f }{\partial z^{\alpha }}} {\rm J}^{\alpha \beta} {\frac{%
\partial g }{\partial z^{\beta }}} \colon = \{ f,g \} \, .$$

(We use the summation convention.) The vector field given by Jd$g$ is called the Hamiltonian vector field generated by $g.$

Casimir invariants are functions that have a zero Poisson bracket with any other function. In other words, given a Poisson bracket, if, for every $g,$ a function $C$ satisfies $\{ g,C \} = 0,$ then $C$ is a Casimir invariant.

This definition is the starting point of any method to compute Casimir invariants. In terms of the tensor J, it becomes $\langle {\rm d}g, {\rm J}%
{\rm d}C \rangle = 0.$ Since $g$ is arbitrary, solving 