When are Vector Fields Hamiltonian?

There is reason to believe that at small scales and low temperatures, quantum mechanical effects will play a role in dissipative systems which arise in solid state physics. References to and a brief discussion of various approaches to the quantisation of the damped harmonic oscillator, can be found in [1]. Although there exist successful and physically intuitive ways to deal with the quantisation of the damped harmonic oscillator, for reasons outlined in [1] we consider a more direct approach to the problem, one which requires an understanding of the Hamiltonian structure of the classical flow. This is why we ask the question from which the article gets its title. Locally it is possible to consider any flow to be Hamiltonian in a sense which we will later make precise. This is true even if the vector field is not conservative. Having understood the Hamiltonian structure of the classical flow, it is possible in principle to *-quantise the system by constructing an associated Moyal-type algebra. This step is based on an insight of Bayen, Flato, Fronsdal Lichnerowicz and Sternheimer [2], who pointed out that the Moyal algebra based on the product