A. Ghose Choudhury

Quantization of the Liénard II equation and Jacobi’s last multiplier

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 Lienard II equation to a position-dependent mass system. The quantization of the Lienard 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.

THE JACOBI LAST MULTIPLIER AND ISOCHRONICITY OF LIÉNARD TYPE SYSTEMS

We present a brief overview of classical isochronous planar differential systems focusing mainly on the second equation of the Lienard type ẍ + f(x)ẋ2 + g(x) = 0. In view of the close relation between Jacobis 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 Veselovs 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.

On commuting vector fields and Darboux functions for planar differential equations

We use Darboux polynomials to obtain an inverse integrating factor and present a method for determining commuting transversal systems for a planar ordinary differential system. We state a result of Garcia and Maza (J. Math. Anal. and Appl. 339 (2008) 740–745) that allows us to construct a linearization starting from a commutator. We illustrate its applicability with two examples. We also construct the rational potential isochrones for planar Hamiltonian systems. In addition we consider the issues of isochronicity and commuting transversal systems from a Lagrangian perspective.

Determination of elementary first integrals of a generalized Raychaudhuri equation by the Darboux integrability method

The Darboux integrability method is particularly useful to determine first integrals of nonplanar autonomous systems of ordinary differential equations, whose associated vector fields are polynomials. In particular, we obtain first integrals for a variant of the generalized Raychaudhuri equation, which has appeared in string inspired modern cosmology.

Symplectic rectification and isochronous Hamiltonian systems

We report the connection of symplectic rectification in the construction of isochronous Hamiltonian systems.

Damped equations of Mathieu type

We obtain the first integrals of various extensions of the Mathieu equation by exploiting the integrable time-dependent classical dynamics introduced by Bartuccelli and Gentile (2003) [6]. We also compute the Lagrangian of the Van der Pol-Mathieu equation using Jacobis last multiplier and consider certain coupled versions of time-dependent equations of the oscillator type.

Application of Jacobi’s last multiplier for construction of Hamiltonians of certain biological systems

The relationship between Jacobi’s last multiplier and the Lagrangian of a second-order ordinary differential equation is quite well known. In this article we demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in biology.

On adjoint symmetry equations, integrating factors and solutions of nonlinear ODEs

We consider the role of the adjoint equation in determining explicit integrating factors and first integrals of nonlinear ODEs. In Chandrasekar et al (2006 J. Math. Phys. 47 023508), the authors have used an extended version of the Prelle–Singer method for a class of nonlinear ODEs of the oscillator type. In particular, we show that their method actually involves finding a solution of the adjoint symmetry equation. Next, we consider a coupled second-order nonlinear ODE system and derive the corresponding coupled adjoint equations. We illustrate how the coupled adjoint equations can be solved to arrive at a first integral.

Canonical Bäcklund transformation for the DST model under open boundary conditions

We study a Backlund transformation for the dimer self-trapping (DST) model under open boundary conditions. As in the periodic case, the transformation is found to be canonical with a corresponding generating function. The spectrality property of the transformation is investigated. Finally, as an application of Backlund transformations we study its connection with discrete-time dynamics.

Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries

Abstract In this paper we compute first integrals of nonlinear ordinary differential equations using the extended Prelle-Singer method, as formulated by Chandrasekar et al in J. Math. Phys. 47 (2), 023508, (2006). We find a new first integral for the Painlevé-Gambier XXII equation. We also derive the first integrals of generalized two-dimensional Kepler system and the Liénard type oscillators.