### The structure of a set of vector fields on Poisson manifolds

Robert I McLachlan
2009 Journal of Physics A: Mathematical and Theoretical
We show that the Lie bracket of an arbitrary vector field with a Hamiltonian vector field is the sum of a Hamiltonian vector field and an energy-preserving vector field, but that not all vector fields can be so decomposed. PACS number: 45.20.Jj Mathematics Subject Classification: 53D17, 37J05 We present an algebraic property of a set of vector fields on a symplectic or Poisson manifold that, while simple, does not appear in the standard sources (e.g. [1, 2] ). Its novel feature is that it
more » ... s non-Hamiltonian and Hamiltonian vector fields. It was discovered in the course of an investigation of series of elementary differentials of a vector field used in geometric numerical integration [3] . Let (P , {, }) be an n-dimensional Poisson manifold and H : P → R a real (C ∞ ) function on P that we call the energy. Let X be the Lie algebra of (C ∞ ) vector fields on P. The two structures {, } and H endow X with a distinguished element, namely the Hamiltonian vector field X H , and with two Lie subalgebras: X Ham , the Lie algebra of Hamiltonian vector fields on P, and X H , the Lie algebra of energy-(i.e. H-) preserving vector fields on P. The Hamiltonian vector field X H lies in both X Ham and X H . Elements of X H are described locally by n − 1 scalar functions, while elements of X Ham are described by single scalar functions. Thus, it makes sense to ask if an arbitrary vector field X (described by n scalar functions) is the sum of a Hamiltonian vector field and an energypreserving vector field. We shall see that this is (i) true locally near regular points of X H , (ii) not necessarily true near singular points of X H and (iii) true globally when X = [Z, X H ] is the Lie bracket of an arbitrary vector field Z with X H . This provides a universal constraint on the range of ad X H . We also have an algebraic description as follows. Proposition 1. [X, X H ] ⊂ X Ham + X H .