1,635 research outputs found
Generating Finite Dimensional Integrable Nonlinear Dynamical Systems
In this article, we present a brief overview of some of the recent progress
made in identifying and generating finite dimensional integrable nonlinear
dynamical systems, exhibiting interesting oscillatory and other solution
properties, including quantum aspects. Particularly we concentrate on Lienard
type nonlinear oscillators and their generalizations and coupled versions.
Specific systems include Mathews-Lakshmanan oscillators, modified Emden
equations, isochronous oscillators and generalizations. Nonstandard Lagrangian
and Hamiltonian formulations of some of these systems are also briefly touched
upon. Nonlocal transformations and linearization aspects are also discussed.Comment: To appear in Eur. Phys. J - ST 222, 665 (2013
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.Comment: 19 page
Extended Prelle-Singer Method and Integrability/Solvability of a Class of Nonlinear th Order Ordinary Differential Equations
We discuss a method of solving order scalar ordinary differential
equations by extending the ideas based on the Prelle-Singer (PS) procedure for
second order ordinary differential equations. We also introduce a novel way of
generating additional integrals of motion from a single integral. We illustrate
the theory for both second and third order equations with suitable examples.
Further, we extend the method to two coupled second order equations and apply
the theory to two-dimensional Kepler problem and deduce the constants of motion
including Runge-Lenz integral.Comment: 18 pages, Article dedicated to Professor F. Calogero on his
70thbirthda
A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case
In this set of papers we formulate a stand alone method to derive maximal
number of linearizing transformations for nonlinear ordinary differential
equations (ODEs) of any order including coupled ones from a knowledge of fewer
number of integrals of motion. The proposed algorithm is simple,
straightforward and efficient and helps to unearth several new types of
linearizing transformations besides the known ones in the literature. To make
our studies systematic we divide our analysis into two parts. In the first part
we confine our investigations to the scalar ODEs and in the second part we
focuss our attention on a system of two coupled second order ODEs. In the case
of scalar ODEs, we consider second and third order nonlinear ODEs in detail and
discuss the method of deriving maximal number of linearizing transformations
irrespective of whether it is local or nonlocal type and illustrate the
underlying theory with suitable examples. As a by-product of this investigation
we unearth a new type of linearizing transformation in third order nonlinear
ODEs. Finally the study is extended to the case of general scalar ODEs. We then
move on to the study of two coupled second order nonlinear ODEs in the next
part and show that the algorithm brings out a wide variety of linearization
transformations. The extraction of maximal number of linearizing
transformations in every case is illustrated with suitable examples.Comment: Accepted for Publication in J. Nonlinear Math. Phys. (2012
On the complete integrability and linearization of nonlinear ordinary differential equations - Part V: Linearization of coupled second order equations
Linearization of coupled second order nonlinear ordinary differential
equations (SNODEs) is one of the open and challenging problems in the theory of
differential equations. In this paper we describe a simple and straightforward
method to derive linearizing transformations for a class of two coupled SNODEs.
Our procedure gives several new types of linearizing transformations of both
invertible and non-invertible kinds. In both the cases we provide algorithms to
derive the general solution of the given SNODE. We illustrate the theory with
potentially important examples.Comment: Accepted for publication in Proc. R. Soc. London
On the complete integrability and linearization of nonlinear ordinary differential equations - Part II: Third order equations
We introduce a method for finding general solutions of third-order nonlinear
differential equations by extending the modified Prelle-Singer method. We
describe a procedure to deduce all the integrals of motion associated with the
given equation so that the general solution follows straightforwardly from
these integrals. The method is illustrated with several examples. Further, we
propose a powerful method of identifying linearizing transformations. The
proposed method not only unifies all the known linearizing transformations
systematically but also introduces a new and generalized linearizing
transformation (GLT). In addition to the above, we provide an algorithm to
invert the nonlocal linearizing transformation. Through this procedure the
general solution for the original nonlinear equation can be obtained from the
solution of the linear ordinary differential equation.Comment: Submitted to Proceedings of the Royal Society London Series A, 21
page
On the complete integrability and linearization of certain second order nonlinear ordinary differential equations
A method of 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. In
addition to the above we introduce a novel way of deriving linearizing
transformations from the first integrals to linearize the second order
nonlinear ordinary differential equations to free particle equation. We
illustrate the theory with several potentially important examples and show that
our procedure is widely applicable.Comment: Proceedings of the Royal Society London Series A (Accepted for
publication) 25 pages, one tabl
- …