On the complete integrability and linearization of nonlinear ordinary differential equations - Part IV: Coupled second order equations

arXiv:0808.0968v2 [nlin.SI] 10 Oct 2008 On the complete integrability and linearization of nonlinear ordinary differential equations - Part IV: Coupled second order equations By V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan Univeristy, Tiruchirapalli - 620 024, India Coupled second order nonlinear differential equations are of fundamental impor- tance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations we focus our attention on the method of deriving general solution of two coupled second order nonlinear ordinary differential equations through the extended Prelle-Singer procedure. We describe a procedure to obtain integrating factors and required number of integrals of motion so that the general solution follows straightforwardly from these integrals. Our method tackles both isotropic and non-isotropic cases in a systematic way. In addition to the above, we introduce a new method of transforming coupled second order nonlinear ODEs into uncoupled ones. We illustrate the theory with potentially important examples. Keywords: Nonlinear differential equations, Coupled second order, Integrability, Integrating factors, Uncoupling 1. Introduction In this part of our study on the integrability and linearization of nonlinear ordinary differential equations (ODEs) we focus our attention on the theoretical formulation and applications of the modified Prelle-Singer (PS) procedure (Prelle & Singer 1983; Duarte et al. 2001; Chandrasekar et al. 2005a; 2006) to a set of two coupled second order ODEs. The need for this demonstration is due to the fact that classifying and studying two degrees of freedom dynamical systems are highly nontrivial problems in the theory of nonlinear dynamical systems. Historically, several techniques have been proposed to identify and obtain general solutions of two coupled second order ODEs. To cite a few we mention Painlev´e analysis, Lie symmetry analysis, general- ized Noether symmetries technique, direct method and so on (Ramani et al. 1989; Lakshmanan & Sahadevan 1993; Bluman & Anco 2002; Lakshmanan & Rajasekar 2003). Each of these methods have their own advantages and limitations. For exam- ple, among the above, certain methods fulfil necessary conditions alone whereas the others guarantee only sufficient conditions for the complete integrability of the sys- tem concerned. This factor alone is a motivating factor to search for more and more powerful methods to isolate and classify integrable and non-integrable dynamical systems. In this direction, by extending the PS procedure and its applications to coupled second order ODEs we argue that the PS method can be used as a stand- Article submitted to Royal Society TEX Paper 2 Chandrasekar, Senthilvelan and Lakshmanan alone technique to solve a wide class of ODEs of any order irrespective of whether it is a single or coupled equation. We mention here that the present analysis is not a straightforward extension of the scalar case. In fact by prolonging the theoretical formulation to the coupled second order ODEs, we deduce the determining equations for the integrating factors and null forms appropriately such that one can obtain the aforementioned functions in a more efficient and straightforward way. Thus the method of obtaining the integrating factors for the given equation is also augmented in this procedure in an efficient manner. Further, while studying the coupled dynamical systems one may face both isotropic and non-isotropic cases. Our method covers both of them in a natural way. In addition to the above, in this paper, we also introduce a new method to transform two coupled second order ODEs to two uncoupled second order ODEs. Thus, the PS procedure inherits several remarkable features both at the theoretical foundations as well as in the range of applications which we have listed out already in Chandrasekar et al. (2005a). Finally, we note that we have carefully fixed the examples so that the basic features associated with this method and the results which it leads to could be explained in an efficient way. The plan of the paper is as follows. In §2 we describe the PS method applicable for coupled second-order ODEs and indicate the new features in finding the inte- grating factors and integrals of motion. In §3 we establish a connection between integrating factors and the form of equations. In §4, the uncoupled equations are briefly considered. In §5, we discuss elaborately the method of constructing integrals and general solutions for the coupled nonlinear ODEs. We support the theory with two nontrivial examples which are being discussed in the contemporary literature. We also briefly discuss the application of our procedure for the case of Liouville integrable systems in §5d. We devote §6 to demonstrate yet another method to identify transformation variables from the first integrals which can be effectively

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