Ja n 20 06 Last multipliers as autonomous solutions of the Liouville equation of transport

Using the characterization of last multipliers as solutions of the Liouville equation of transport, new results in this approach of ODE are given by obtaining several new characterizations e.g. in terms of Witten and Marsden differentials. Applications to Hamiltonian vector fields on Poisson manifolds and vector fields on Riemannian manifolds are presented. In the Poisson case the unimodular bracket gives a major simplification in computations while in the Riemannian framework a Helmholtz type decomposition yields three remarkable examples: one is the quadratic porous medium equation, the second (the autonomous version of previous) produces harmonic square functions while the last is about the gradient of a the distance function with respect to a 2D rotationally symmetric metric. 2000 Math. Subject Classification: 58A15; 58A30; 34A26; 34C40.