Symmetries of Systems with the Same Jacobi Multiplier

Abstract

The concept of the Jacobi multiplier for ordinary differential equations up to the second order is reviewed and its connection with classical methods of canonical variables and differential invariants is established. We express, for equations of the second order, the Jacobi multiplier in terms of integrating factors for reduced equations of the first order. We also investigate, from a symmetry point of view, how two different systems with the same Jacobi multiplier are interrelated. As a result, we determine the conditions when such systems admit the same two-dimensional Lie algebra of symmetries. Several illustrative examples are given.