this equation amounts to finding the functions $C$ from the condition that the vector field ${\rm J%
}{\rm d}C$ vanishes. In the case of a finite-dimensional system, this computation requires solving a set of coupled partial differential equations for the function $C$: ${\rm J}^{ij} \partial C / \partial z^j =0$. However, a more efficient method for computing Casimir invariants follows from an alternate geometric interpretation of $\{ g,C \}=0.$ Making use of the antisymmetry of the Poisson bracket, we rewrite $\langle {\rm d}g,{\rm J}%
{\rm d}C \rangle = 0$ as $\langle {\rm d}C,{\rm J}{\rm d}g \rangle = 0.$ Since this equation must hold for arbitrary $g$, we can interpret it as saying that the differential one-form given by ${\rm d}C$ is annulled by every vector field in the image of J.

Geometrically, this means that the vector field Jd$g$ must lie everywhere tangent to the level set of the function $C$. In this case, any integral curve of this (or any other) Hamiltonian vector field must lie within a single level set of $C$, which is a way of saying that $C$ is a constant of motion.

These considerations help us to compute Casimir invariants in the following way. The 