The Jacobi metric derived from the line element by one of the authors is shown to reduce to the standard formulation in the non-relativistic approximation. We obtain the Jacobi metric for various stationary metrics. Finally, the Jacobi-Maupertuis metric is formulated for time-dependent metrics by including the Eisenhart-Duval lift, known as the Jacobi-Eisenhart metric.

In this paper, we return to the subject of Jacobi metrics for timelike and null geodesics in stationary spacetimes, correcting some previous misconceptions. We show that not only null geodesics but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulas. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally, we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behavior of Maxwell’s equations in the curved spacetime. In particular, we see in very concrete terms how the gravitational electric permittivity, magnetic permeability, and magnetoelectric susceptibility enter the Jacobi-Maupertuis Randers-Finsler function.

Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Liénard equation. By combining the Chiellini condition for integrability and Jacobi's Last Multiplier the Lagrangian and Hamiltonian of the Liénard equation is derived. We also show that the Kukles equation is the only equation in the Liénard family which satisfies both the Chiellini integrability and the Sabatini criterion for isochronicity conditions. In addition we examine this result by mapping the Liénard equation to a harmonic oscillator equation using tacitly Chiellini's condition. Finally we provide a metriplectic and complex Hamiltonian formulation of the Liénard equation through the use of Chiellini condition for integrability.