M. Senthilvelan

On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations

A method for finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle–Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. We also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second-order nonlinear ordinary differential equations to free particle equations. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators

A non-linear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. This model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the non-linear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this non-linear system to the harmonic oscillator on spaces of constant curvature, the two-dimensional sphere S2 and hyperbolic plane H2, is discussed.

A simple and unified approach to identify integrable nonlinear oscillators and systems

In this paper, we consider a generalized second-order nonlinear ordinary differential equation (ODE) of the form x+(k1xq+k2)x+k3x2q+1+k4xq+1+λ1x=0, where ki’s, i=1,2,3,4, λ1, and q are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing–van der Pol oscillators, modified Emden-type equation and its hierarchy, generalized Duffing–van der Pol oscillator equation hierarchy, and so on, and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent q,q∊R. The q=1 and q=2 cases are analyzed in detail and the results are generalized to arbitrary q. Our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature. T...

On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator

Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

On the general solution for the modified Emden-type equation

In this paper, we demonstrate that the modified Emden type equation (MEE),

A unification in the theory of linearization of second-order nonlinear ordinary differential equations

\ddot{x}+\alpha x\dot{x}+\beta x^3=0

Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator

, is integrable either explicitly or by quadrature for any value of

On the complete integrability and linearization of nonlinear ordinary differential equations. V. Linearization of coupled second-order equations

\alpha

A group theoretical identification of integrable equations in the Liénard-type equation x+f(x)x+g(x)=0. II. Equations having maximal Lie point symmetries

and

Nonstandard conserved Hamiltonian structures in dissipative/damped systems: Nonlinear generalizations of damped harmonic oscillator

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