Holomorphic last multipliers on complex manifolds

Abstract

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic last multipliers with real last multipliers. Also, we consider some applications of our study for holomorphic gradient vector fields on holomorphic Riemannain manifolds as well as for holomorphic Hamiltonian vector fields and holomorphic Poisson bivector fields on holomorphic Poisson manifolds.

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

“…As the theory of Jacobi multipliers is not well known in physics, in spite of its many applications, the aim of this paper is to give a short updated review of this concept from a geometric perspective (see also [31][32][33]) and show by means of illustrative examples some of its physical applications. The concept of Jacobi multipliers is introduced in Section 2, where the particular case of a two-dimensional autonomous system of first-order ordinary differential equations illustrates the theory.…”