Mechanics, Materials Science & Engineering, July 2016
ISSN 2412-5954
Lagrangian Representations and Solutions of Modified Emden-Type Equations Aparna Saha1 and Benoy Talukdar1
1
Department of Physics, Visva-Bharati University, Santiniketan 731235, India DOI 10.13140/RG.2.1.2683.0323
Keywords: variation of parameters, factorization method, modified Emden-type equations, solutions, Lagrangian representation.
ABSTRACT. We derive two novel methods to construct solutions of the physically important modified Emden-type ethod of variation of parameters, we make use of a trivial particular solution of the problem to construct expressions for the nontrivial general solutions. Secondly, we judiciously adapt the factorization method of differential operators to present solutions of certain oscillator equations obtained by adding a linear term to the MEEs. We provide Lagrangian and Hamiltonian formulations of these equations in order to look for another useful theoretical model for solving MEEs.
PACS numbers: 02.30.Hq, 02.30.Ik, 05.45.-a
1. Introduction. The modified Emden-type equation [1] is given by:
,
where .
and
,
(1)
are arbitrary parameters, and overdots denote differentiation with respect to time
This equation plays a role in several applicative contexts including the study of equilibrium configurations of a spherical gas cloud with mutual interaction between the molecules subject to laws of thermodynamics [2]. An important mathematical observation on (1) is that only for it possess eight parameter Lie point symmetries [3] and is exactly solvable. In the recent past Chandrasekar, Senthilvelan and Lakshmanan [4] made use of an extension of the semi-decision algorithmic method [5] supplemented by a suitable Hamiltonian theory to demonstrate that (1) is integrable for arbitrary values of and . Iacono [6] proved the integrability of (1) by mapping it nonlinear differential equations once their forms are found by a specific method [7]. However, we note that, as opposed to the theory of linear differential equations, the nonlinear theory is highly fragmented. Thus the identification of a new mathematical procedure and demonstration of its usefulness in solving nonlinear equations will always remain interesting provided the method of solution is uncomplicated enough to be appreciated by a wide variety of audience. The present work is an effort in this direction. In particular, we shall present two independent methods to solve MEEs one b suitable factorization of the differential operators [1] that give the equations. To the best of our knowledge these methods have not hitherto been discussed in the literature. In addition, we provide MMSE Journal. Open Access www.mmse.xyz
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