The Jacobi last multiplier and isochronicity of Liénard type systems (Q2846472)

A classical dynamical system is said to be isochronous if it displays an open region in its phase space in which all its solutions are completely periodic (periodic in all degrees of freedom) with the same fixed period. The linear harmonic oscillator is the prototype of an isochronous system and all other isochronous systems are isoperiodic with the harmonic oscillator. Moreover, there are only two rational potentials which can support isochronous motion, namely, the harmonic oscillator and the isotonic potential. The latter is a harmonic oscillator potential with a centripetal barrier. In fact, any potential derived from \(V=\frac{1}{2} \omega^2 x^2\) by shearing is an isotonic potential. In the quantum context the problem of isochronicity essentially consists in finding those potentials having an equispaced energy spectrum. Here, also the harmonic oscillator and the isotonic oscillator are the two main examples of such systems.NEWLINENEWLINEIn the paper the authors present a brief overview of classical isochronous planar differential systems focusing mainly on the second-order equation of the Liénard type NEWLINE\[NEWLINE\ddot{x}+f(x) \dot{x}^2 + g(x)=0.NEWLINE\]NEWLINE In previous works of the authors, a Lagrangian is constructed for the Liénard type system using the Jacobi last multiplier (JLM) together with the corresponding Hamiltonian function. Constructing an obvious change of variables, the authors could map the Hamiltonian to the linear harmonic oscillator or to the isotonic potential, which permits to study isochronous systems using the JLM. Hence, in view of the close relation between Jacobi's last multiplier and the Lagrangian of such a second-order ordinary differential equation, it is possible to assign a suitable potential function to this equation. Using this along with Chalykh's and Veselov's result regarding the existence of only two rational potentials which can give rise to isochronous motions for planar systems, the authors attempt to clarify some of the previous notions and results concerning the issue of isochronous motions for this class of differential equations. In particular, they provide a justification for the Urabe criterion giving a derivation of the Bolotin-MacKay potential. The method is illustrated with several well-known examples like the quadratic Loud system and the Cherkas system.