Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

SS symmetry Article Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods Muhammad Safdar 1, *, Asghar Qadir 2 and Muhammad Umar Farooq 3 1 2 3 * School of Mechanical and Manufacturing Engineering (SMME), National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan School of Natural Sciences (SNS),National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan; Department of Basic Sciences & Humanities, College of E & ME, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan; Correspondence: Received: 27 June 2019; Accepted: 19 August 2019; Published: 17 September 2019



 Abstract: Noether symmetries and first integrals of a class of two-dimensional systems of second order ordinary differential equations (ODEs) are investigated using real and complex methods. We show that first integrals of systems of two second order ODEs derived by the complex Noether approach cannot be obtained by the real methods. Furthermore, it is proved that a complex method can be extended to larger systems and higher order. Keywords: systems of ODEs; Noether operators; Noether symmetries; first integrals 1. Introduction Lie developed a symmetry method for solving differential equations (DEs) [1–4]. Noether [5] used these methods to prove that, for DEs obtained from a variational principle, for each symmetry generator there is a corresponding invariant, first integral. These symmetries are called Noether and, if they exist, then Noether’s theorem readily provides the associated first integrals. Since they provide a double reduction of the order of the equation, and a sufficient number can actually be used to solve the equation, it is worthwhile to obtain them. Furthermore, they are useful for studying the physical aspects of the dynamical systems, like time translational symmetry gives energy conservation, spatial translation provides momentum conservation and rotational symmetry implies conservation of angular momentum. For a scalar ODE, the corresponding Lagrangian has a five-dimensional maximal Noether symmetry algebra, as guaranteed by a theorem [6], and all the lower dimensions (obviously except 4). Though Lie methods involved complex functions of complex variables, they did not make explicit use of the Cauchy–Riemann (CR) equations. These conditions provide an auxiliary system of DEs satisfied by the corresponding system of DEs obtained by splitting the complex functions of the scalar or systems of DEs into the two real ones. One obtains either a system of partial differential equations (PDEs), if the independent variable is complex or a system of ODEs if it is real. The explicit use of complex functions of complex or real variables is demonstrated in [7–10] where solvability of systems of DEs is achieved through Noether symmetries and corresponding first integrals. Furthermore, by employing complex symmetry procedures: the energy stored in the field of a coupled harmonic oscillator was studied in [11] and linearizability of systems of two second order ODEs was addressed Symmetry 2019, 11, 1180; doi:10.3390/sym11091180 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1180 2 of 12 in [12,13]. The complex procedure, indeed, has been extended to higher dimensional systems of second order ODEs [14] and two-dimensional, systems of third order ODEs [15]. In this paper, we extend the use of complex symmetry methods further to obtain invariants of systems of ODEs and demonstrate that we can obtain new invariants not obtainable by the usual, non-complex, methods. The new invariants for systems arise due to complex Lagrangians and first integrals of the base ODEs involving complex dependent functions of the real independent variables. Complex symmetries have already been used to construct first integrals through Noether symmetries and derive invariants for two-dimensional, systems of second order ODEs [8–10]. We first compare the usual (real) and complex Noether approaches developed to derive first integrals for systems of two second order ODEs. We find that the latter yields more first integrals than the former for these systems. The first integrals derived using a complex procedure also satisfy the conditions of the real Noether’s theorem that exists for systems of ODEs. Next, we prove that Lagrangians and corresponding first integrals of the complex scalar ODEs will always split into two real Lagrangians and first integrals for the corresponding system of two equations. For this purpose, we use the CR-equations, which are satisfied by the Lagrangians and first integrals provided by the complex procedure. Furthermore, we show that the complex Noether symmetries do not, in general, split into two Noether symmetries of the corresponding systems. The thrust is not to find directly applicable invariants, which could turn up but to demonstrate how a complex method can provide new invariants and insights into Noether symmetries and first integrals. This work also suggests that the class of systems presented here should, indeed, be singled out when classifying systems of ODEs on the basis of their Noether symmetries and first integrals as it may not follow the classifications presented by employing real symmetry methods. Theorems and their proofs in the later part of this paper show that the method adopted here can trivially be extended to higher dimensions and order of ODEs. The plan of the paper is as follows: the next section gives the procedures to derive Noether symmetries, operators and corresponding first integrals for systems of two second order ODEs. In the third section, we obtain Noether symmetries and first integrals for two-dimensional, systems of second order ODEs using a real symmetry method. In the subsequent section, first integrals for these systems are derived by employing complex procedures. We end with a concluding section, which also gives the proofs of the claims given in the previous section. 2. Preliminaries For a system of two coupled (in general) nonlinear ODEs y00 = S1 ( x, y, z, y0 , z0 ), z00 = S2 ( x, y, z, y0 , z0 ), (1) where prime denotes derivative with respect to x, and the point symmetry generator is X = ξ ( x, y, z)∂ x + η1 ( x, y, z)∂y + η2 ( x, y, z)∂z , (2) where ξ, η1 , and η2 , are the functions that appear in the infinitesimal coordinate transformations of the dependent and independent variables, ∂ x = ∂/∂x, etc. The first extension of X is X [1]  = X+ d d η − y0 ξ dx 1 dx   ∂y0 + d d η2 − z 0 ξ dx dx  ∂z0 , (3) where d/dx = ∂ x + y0 ∂y + z0 ∂z + · · · . If system (1) admits a Lagrangian L( x, y, z, y0 , z0 ), then it is equivalent to the Euler–Lagrange equations d dx  ∂L ∂y0  ∂L − = 0, ∂y d dx  ∂L ∂z0  − ∂L = 0. ∂z (4) Symmetry 2019, 11, 1180 3 of 12 The vector field (2) is called a Noether symmetry generator corresp (...truncated)