Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Demina, Maria V., Nechitailo, Varvara G.
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Russian Science Foundation
Award: G1530772311
OpenAlex ID: W7223223577