Generalized poisson mechanics in D-branes

Following the work of David Fairlie and his collaborators, we discuss the geometry of the equations of motion for the companion Lagrangian. This arises while we attempt to construct the field theoretic Lagrangians for strings and branes. These equations are related to generalized Bateman equations. Interestingly, an inhomogeneous form of these equations are related to what Dubrovin and Novikov called equations of the hydrodynamic type. We translate this set of equations into (Nambu)-Poisson equations and formulate the Lax representation of these systems. We deform all this set of equations by the Moyal product. We also present the Lax pair of the Moyal deformed systems for some special cases.