II. NON-HAMILTONIAN BRACKETS AND MEASURE-PRESERVING ALGO- RITHMS Consider an arbitrary system admitting a time-independent (extended) Hamiltonian ex- pressed in terms of the phase space coordinates x$_{i}$ , i = 1 , . . . , 2 N . In this case, the Hamil- tonian can be interpreted as the conserved energy of the system. Upon introducing an antisymmetric tensor field (generalized Poisson tensor [21, 33]) in phase space, B ( x ) = − B $^{T}$( x ), one can define non-Hamiltonian brackets [14–16] as where a = a ( x ) and b = b ( x ) are two arbitrary phase space functions. The bracket defined in Eq. (1) is classified as non-Hamiltonian [14–16] since, in general, it does not obey the Jacobi relation, i.e., in general the Jacobiator J ̸ = 0, where [21] with c = c ( x ) arbitrary phase space function (in addition to the functions a and b , previously introduced). If J ̸ = 0, the tensor B$_{ij}$ is said to define an ‘almost-Poisson’ structure [34]. (Such systems have also been called ‘pseudo-Hamiltonian’ [33].) An energy-conserving and in general non-Hamiltonian phase space flow is then defined by the vector field where conservation of H ( x ) follows directly from the antisymmetry of B$_{ij}$ . It has previously been shown how equilibrium statistical mechanics can be comprehen- sively formulated within this framework [16]. It is also possible to recast the above formalism and the corresponding statistical mechanics in the language of differential forms [17, 18]. If the matrix B is invertible (this is true for all the cases considered here), with inverse Ω$_{ij}$, we can define the 2-form [35] The dynamics of Eq. (3) is then Hamiltonian if and only if the form (4) is closed , i.e., has zero exterior derivative, d Ω = 0 [35]. This condition is independent of the particular system of coordinates used to describe the dynamics. { a, b } = 2 n ∑ i,j =1 ∂a ∂x$_{i}$ B$_{ij}$ ∂b ∂x$_{j}$ , (1) J = { a, { b, c }} + { b, { c, a }} + { c, { a, b }} , (2) ˙ x$_{i}$ = { x$_{i}$, H } = 2 N ∑ j =1 B$_{ij}$ ∂H ∂x$_{j}$ , (3) Ω = 1 $_{2}$Ω$_{ij}$ dx i ∧ dx $^{j}$. (4) 4