an arbitrary function.

Note the distinct origins of the two types of functional dependence $K$ in these invariants. Its dependence on $f$ arises from the structure of the infinite-dimensional bracket alone, and so is common to any kinetic theory to which bracket (\[INFBRAC\]) is applicable: it might be called the “macroscopic” part of the Casimir invariant. On the other hand, the dependence of $K $ on the $C^{(i)}$ arises from the structure of the finite-dimensional inner bracket, and so reflects the nature of the “microscopic” theory.

Conclusion {#S-CONCL}
==========

We have presented a method for computing Casimir invariants applicable to both finite and infinite-dimensional Poisson brackets. We believe that presenting this method geometrically has two benefits: it enhances pictorial intuition of the behavior of finite-dimensional Hamiltonian systems, and it gives insight into how infinite-dimensional Hamiltonian systems can be treated analogously to finite-dimensional ones.

We have also computed some new examples of Casimir invariants. Most interesting, perhaps, are those that are the results of Poisson structures at finite and infinite-dimensional levels simultaneously. In these, we have pointed out what parts of the invariants correspond to which Poisson structure.

[**Acknowledgements**]{}

BH acknowledges Comunidad Autónoma de Madrid for financial support, as well as 