Dissipative-Hamiltonian decomposition of smooth vector fields based on symmetries

We propose a method to decompose a smooth vector field into conservative and dissipative components. The procedure is based on the identification of the kernel of a linear operator associated with a given Hamiltonian combined with the use of Lie transformations for vector fields. Moreover, under certain conditions the nonconservative part of the splitting can be dropped at a given order of the transformation, obtaining after truncation, a Hamilton vector field. The technique is illustrated through the application to the motion of a particle subject to the potential of a champagne bottle plus a small friction.