3,842 research outputs found
On the Linearization of Second-Order Differential and Difference Equations
This article complements recent results of the papers [J. Math. Phys. 41
(2000), 480; 45 (2004), 336] on the symmetry classification of second-order
ordinary difference equations and meshes, as well as the Lagrangian formalism
and Noether-type integration technique. It turned out that there exist
nonlinear superposition principles for solutions of special second-order
ordinary difference equations which possess Lie group symmetries. This
superposition springs from the linearization of second-order ordinary
difference equations by means of non-point transformations which act
simultaneously on equations and meshes. These transformations become some sort
of contact transformations in the continuous limit.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Linearization of nonlinear connections on vector and affine bundles, and some applications
A linear connection is associated to a nonlinear connection on a vector
bundle by a linearization procedure. Our definition is intrinsic in terms of
vector fields on the bundle. For a connection on an affine bundle our procedure
can be applied after homogenization and restriction. Several applications in
Classical Mechanics are provided
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
Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations
The linearization problem of a second-order ordinary differential equation by
the generalized Sundman transformation was considered earlier by Duarte,
Moreira and Santos using the Laguerre form. The results obtained in the present
paper demonstrate that their solution of the linearization problem for a
second-order ordinary differential equation via the generalized Sundman
transformation is not complete. We also give examples which show that the
Laguerre form is not sufficient for the linearization problem via the
generalized Sundman transformation
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