Jacobi last multipliers and metrisability of Liénard differential equations

Abstract

We classify non-autonomous Liénard differential equations possessing invariants quadratic with respect to the first derivative with cofactors of a special form. These invariants produce Jacobi last multipliers of the related equations. We find some families of Liénard differential equations that have first integrals fractional-quadratic with respect to the first derivative. Our method can be used to solve the metrisability problem for Liénard differential equations. A second-order ordinary differential equation is called metrisable if its integral curves are unparameterized geodesics of a Levi-Civita connection of a metric on a two-dimensional surface. We present some explicit examples. We find metrisable autonomous Liénard differential equations satisfying the Chiellini integrability condition. This result gives metrics with rational or transcendental in momenta first integrals. We derive a positive-definite metric whose geodesic flow possesses a first integral that is non-Liouvillian in the momentum ratio. We prove that the Ermakov-Pinney equation is metrisable.