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On the Reduction Process of Nucci-Reduce Algorithm for Computing Nonlocal Symmetries of Dynamical Systems:A case study of the Kepler and Kepler-related problems

The snags in Nucci(1996)REDUCE algorithm are the intrinsic computational efforts and the ability to recognize the ignorable variable(s) during the reduction process of the algorithm. An inappropriate choice of the ignorable variable(s)may lead to an infinite loop. We construct an isomorphic transformation which ameliorates these problems, and with which a simple, definite steps of algebraic process, produced equivalent system of equations to that of Nucci that are easily solved by Lie point symmetry algorithm.


INTRODUCTION
In 1994 Krause used the Kepler problem as a vehicle for the introduction of his concept of the complete symmetry group of a differential equation or systems of differential equations. Essentially the complete symmetry group of a differential equation is the group of the algebra of the Lie symmetries required to specify completely the differential equation [2,1]. In the case for the Kepler problem, the group comprises of nonlocal symmetries although it is not the case for every differential equations [1,12,5]. These nonlocal symmetries are not Noether symmetries (Variational symmetry).
Since Krause (1994) the concern for nonlocal symmetry became prominent in the domain of symmetry analysis of dynamical systems [11,5,13,8]. In 1996, Nucci unveil a profound algorithm for computing the nonlocal symmetries of the Kepler problem where she combined her reduce technique with the Lie point algorithm, and thereby invalidated Krause (1994) assertion that the complete symmetry group of the Kepler problem could not be ascertained by the Lie group analysis. Since then a substantial literature is devoted to topics in nonlocal symmetry analysis. In this regards the NUCCI-REDUCE algorithm software is acclaimed the most effective and efficient of all the techniques available in the literature to date [11,2,4,3]. However, attempting to use this REDUCE algorithm to compute the non-trivial problems with pencil and papers could be very complicated and complex. It suffices to say that one must develop ingenuity in the choice of change of variables and computational skills to reduce the nonlinear system to a system of two linear equations with one second order and ) 1 ( − n first order (for an n-dimensional argument) equations which we now apply the Lie group algorithm to obtain the point symmetries in the final new variables. These are transformed back to the original variables to obtain both point and nonlocal symmetries of the original problem. The Nucci REDUCE algorithm had been used successfully to compute both types of symmetries for all other problems possessing conserved vectors of the Laplace-Runge-Lenz and Poincaré type, and even for Lagrangian systems and Hamiltonian equations [7,12,11,6]. It is also true that the Nucci-reduce algorithm is successfully applicable to any system of n first-order equations, which are transformable into equivalent systems where at least one of the equations is of the second order and the admitted Lie symmetry algebra is no longer infinite dimensional and the nontrivial symmetries of the original system could be retrieved [9,3,14]. This paper is organized in the following manner; firstly it recalled the Nucci reduction algorithm with associated deficiencies in sections 1.0 and 2.0, followed by the reduction The radial and transverse components of the motion are as follows The algorithm yields the following equations for the i w : Using the same procedure above to eliminate 3 w , we obtain So, we conveniently choose and equation (2.6) and (2.8) reduced to the system As in the Kepler problem, this system also reduced to a harmonic oscillator and a conservation law. This similarity between the two systems (1.1) and (2.1) has already been reported [13].We note that where the equation of motion under consideration using Nucci reduction process has more than two independent variables, the reduction process become cumbersome [7]. The cases of the Kepler problem in 3-dimensions and the MICZ problem are simple acid test cases. Nucci algorithm involved a set of six first order equations which are further transformed into five equations (one second order and four first order) [11,12]. Then we required five new and a set of three variables from which a choice of ignorable variable is made. Further, we are required to eliminate from a set of four first order equations a certain variable to be able to obtain the z-component of the parabolic determining equation which variables provide new dependent variables that reduce the system to two second order equations and a conservation law. The expressions involved in these processes are not simple. It is an uphill task to derive the first derivatives of the variables involved with pencil and papers [7]. Consequently, the reduction process required high level of symbolic computer manipulations code that are computer time consuming.
The algorithm to find first integrals which are not necessarily "physical" conserved quantities, suffice it to say that there is need to adjust the algorithm of the reduction of order introduced by Nucci since all symmetries are found by Lie point method, and there exist natural variables with which to reduce the equation of motion to suitable system of linear equations and conservation laws [13,6].

Our proposed reduction process
We construct an isomorphic transformation ) such that the transformation is invariant, respect the constants of the motion, and preserved the orbit of the motion.

Equation (1.2) is rewritten as
With second prolongation given by (3.8) Hence the radial equation becomes  The pair of equations corresponding to (6.5) is This system also possesses nine symmetry generators. However the explicit expressions for the generators are much more complicated and we do not proceed further in this case.

6.1
The symmetry group of the MICZ on the cone is obtained by the following transformation: symmetry becomes a ) , , ( 2 1 u u φ symmetry according to following and by considering the relations The representations of the complete symmetry group of the Kepler problem abound in the literature. If in the symmetry representation of the Kepler problem [8], we replace 2 L by 2 2 λ L + we obtain the complete symmetry group representation of the MICZ problem from (6.8) and (6.9) on its cone of motion as following except 1 Λ which is a completely different from the scaling symmetry from that of the Kepler problem: and note that for 0 = λ , we have the Kepler problem.
In the Kepler case, the two nonlocal symmetries to be included to form the complete symmetry group are of type ± 4 Λ . We have not seen where this has been reported.

Concluding remarks
It is common knowledge that all point symmetries can be determined by Lie group algorithm; there is also the potential for a loss of symmetries in the reduction process just as there is hope for an increase in the total number of symmetries [13]. The algorithm for translating back to symmetries in the original variables of the unreduced system is not ambiguous as it where [13]. Sine-qua-non, the concerns of the reduction process ought to be more centered on reducing the system to a set of equations which one can easily apply Lie algorithm to, rather than involving extraneous ambiguities such as determining first integrals and complexities of the choice of varying variables that involved complex computer manipulations and computer time wastage.
In this paper we have been able to establish with examples a simple algebraic process for reducing systems to equivalent systems which Nucci algorithm could obtain that are easily solved for their symmetries by Lie group analysis without ambiguities. We report here that we have used this algebraic process to reduce other nonlinear systems to convenient forms which we successfully applied the Lie algorithm to and we obtained the Lie point symmetries and subsequently obtained the nonlocal symmetries in the original systems, both in two and three dimensional coordinate systems. To mention but a few, the Kepler-Ermakov, Ermakov-Toy, and any physical problem possessing the Poincaré vector are examples that we have reduced with this algebraic process.
The analysis of their symmetry algebras of equations of motion on the cone is a subject for discussion in another paper.