# Why are Hamiltonian's used with non-Hamiltonian brackets to define equations of motion?

I am reading Non-Hamiltonian equilibrium statistical mechanics and I am confused about how they are using Hamiltonians.

In Section II, they introduce the "non-Hamiltonian bracket", \begin{align} \{a, b\} &= \sum_{ij}^{2N} \frac{\partial a}{\partial x_i} B_{ij} \frac{\partial b}{\partial x_j}, \tag{1} \end{align} where $B_{ij} = -B_{ji}$ and $x = (q, p)$.

Then they define, \begin{align} \dot{x}_k = \{ x_k, \mathcal{H} \}. \tag{2} \end{align}

Where did this Hamiltonian come from? We are working with non-Hamiltonian systems, how are we defining a Hamiltonian? Why do I need a Hamiltonian for non-hamiltonian dynamics?

• Unfortunately, not all knowledge is open access yet, so it's hard to tell if one cannot look at the definitions from the cited papers. My guess is that the idea is to use a non-Hamiltonian bracket to generate equations in the same fashion as in Hamiltonian mechanics. I.e., take a function $\mathcal H$ and stick it into the bracket as in (2). Interpret these as equations of motion. Given that the simplectic form is not the canonical one in general, the equations are "non-Hamiltonian" (in general). – Phoenix87 Oct 17 '17 at 21:54
• @Phoenix87 Oh. I didn't realize that the term "Hamiltonian" didn't cover symplectic forms that are not canonical. That makes a lot of sense because the whole paper is about manipulating $\mathcal{H}$ the non-Hamiltonian – aidan.plenert.macdonald Oct 18 '17 at 0:21