We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé-Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of secondorder ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.

A dilute bose gas in a quasi two-dimensional harmonic trap and interacting with a repulsive two-body zero-range potential of fixed coupling constant is considered. Using the Thomas-Fermi method, it is shown to remain in the same uncondensed phase as the temperature is lowered. Its density profile and energy are identical to that of an ideal gas obeying the fractional exclusion statistics of Haldane.

In this paper, the role of Jacobi’s last multiplier in mechanical systems with a position-dependent mass is unveiled. In particular, we map the Liénard II equation to a position-dependent mass system. The quantization of the Liénard II equation is then carried out using the point canonical transformation method together with the Von Roos ordering technique. Finally, we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions.

We consider a large class of polynomial planar differential equations proposed by Cherkas (1976 Differensial'nye Uravneniya 12 201–6), and show that these systems admit a Lagrangian description via the Jacobi last multiplier (JLM). It is shown how the potential term can be mapped either to a linear harmonic oscillator potential or into an isotonic potential for specific values of the coefficients of the polynomials. This enables the identification of the specific cases of isochronous motion without making use of the computational procedure suggested by Hill et al (2007 Nonlinear Anal.: Theor. Methods Appl. 67 52–69), based on the Pleshkan algorithm. Finally, we obtain a Lagrangian description and perform a similar analysis for a cubic system to illustrate the applicability of this procedure based on Jacobi's last multiplier.

We examine the analytical structure of the nonlinear Liénard oscillator and show that it is a bi-Hamiltonain system depending upon the choice of the coupling parameters. While one has been recently studied in the context of a quantized momentumdependent mass system, the other Hamiltonian also reflects a similar feature in the mass function and also depicts an isotonic character. We solve for such a Hamitonian and give the complete solution in terms of a confluent hypergeometric function.Mathematical Classification 70H03, 81Q05. *

We present a brief overview of classical isochronous planar differential systems focusing mainly on the second equation of the Liénard typeẍ + f (x)ẋ 2 + g(x) = 0. 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 and Veselov's result regarding the existence of only two rational potentials which can give rise to isochronous motions for planar systems, we attempt to clarify some of the previous notions and results concerning the issue of isochronous motions for this class of differential equations. In particular, we provide a justification for the Urabe criterion besides giving a derivation of the Bolotin-MacKay potential. The method as formulated here is illustrated with several well-known examples like the quadratic Loud system and the Cherkas system and does not require any computation relying only on the standard techniques familiar to most physicists.

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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. Mathematics Classification (2010) :