Variational principle and phase space measure in non-canonical coordinates

Theoretical formalisms very often use non-canonical equations of motion. For example, the equations for Eulerian variables, that describe ideal continuous media, are in general non-canonical [1]. Non-canonical phase space flows can be derived from Hamiltonian dynamics by means of non-canonical transformations of phase space coordinates (i.e. transformations with Jacobian not equal to one) while non-Hamiltonian dynamics cannot be derived using only transformations of phase space coordinates. However, non-canonical dynamics has a certain likeness with energy-conserving non-Hamiltonian dynamics [2] (this latter is commonly used in molecular dynamics simulations). For this reason, non-canonical systems can be used to improve our understanding of non-Hamiltonian systems with a conserved energy [2]. In the following, the comparison between non-Hamiltonian and non-canonical system will be exploited to clarify some issues regarding phase space measure. It will be shown that the invariant measure of non-canonical phase space can be derived from coordinate transformations without the need to consider dynamical properties, such as phase space compressibility. The situation is different in the non-Hamiltonian case, where one has to resort to arguments associated with time evolution in phase space [2, 3]. Coordinate transformations will be also used to obtain the covariant form of the entropy functional and a variational principle for non-canonical equations of motion, arising from a symplectic form of the action.