Solving regularized equations in the form of polynomials

: The article is devoted to the determination of second-order perturbations in rectangular coordinates and components of the body motion to be under study. The main difficulty in solving this problem was the choice of a system of differential equations of perturbed motion, the coefficients of the projections of the perturbing acceleration are entire functions with respect to the independent regularizing variable. This circumstance allows constructing a unified algorithm for determining perturbations of the second and higher order in the form of finite polynomials with respect to some regularizing variables that are selected at each stage of approximation. The number of approximations is determined by the given accuracy. It is rigorously proven that the introduction of a new regularizing variable provides a representation of the right-hand sides of the system of differential equations of perturbed motion by finite polynomials. Special points are used to reduce the degree of approximating polynomials, as well as to choose regularizing variables.


Introduction
One of the crucial tasks associated with trajectory measurements is the determination of the partial derivatives of rectangular coordinates that make up the body motion speed with respect to the initial conditions. In operations [1][2] added auxiliary functions, which are degree series with respect to the auxiliary variable. In operations [3][4][5] outlined ways of using universal variables in a number of tasks of mechanics to determine disturbances by the method of variation of arbitrary constants. In this case, it is convenient to consider the components of the initial values of the radius-vector and velocity as osculating variables. New methods for determining disturbances keep the standard features of the classical ones, while calculating the disturbances, the small parameter method is used, which makes it possible to obtain asymptotic decomposition of the solution. Recently, Picard method for integration of differential equations is more commonly used, which leads to a convergent process of successive approximations that gives a solution to a system of differential equations. The error of the solution depends on the accuracy of the initial approximation of the perturbation function. General principles of the development of perturbation theory in coordinates were analyzed in operations [6][7][8] studied the use of regularizing variables for calculation of trajectories of motion in operation. The results of this research show that the use of regularizing variables increases the computer-based accuracy of calculations and significantly reduces the calculation time. A crucial task of mechanics is to approximate the rectangular coordinates that make up the body speed and time in case of disturbed motion by algebraic polynomials of the lowest degree with respect to the auxiliary variable with a predetermined degree of accuracy.
One of the important problems in mechanics is the approximation of rectangular coordinates constituting the body velocity and time when the motion is perturbed by the lowest degree algebraic polynomials relative to the auxiliary variable with a predetermined degree of accuracy. This paper describes a special system of differential equations of the perturbed moving body and this system is integrated through successive approximations method, which using the coordinates and constituents body velocity, take the form of polynomials in powers of some auxiliary variable. Its own independent variable is taken at each approximation step.

Mathematical model
In articles [9][10][11], there were obtained rectangular coordinates x,y regularized speed components x', y' time t as an initial approximation in the following form: (1) First order perturbations in auxiliary quantities are defined as follows [11] ( ), ( ), ( ) n n n P w P w P w  polynomials in powers of the regularizing variable w to some degree n,  (4), (3) into the expressions for the rectangular coordinates and time given in article [11], we obtain By performing the indicated replacement of an independent variable w, we obtain that the sought integral of function (6) has the form: where ( ) n R shv  a polynomial function in powers of shv.
The integral (8) has the following structure where 0 1 2 , , l l l  some factors, ( , ) Q shv chv   a polynomial function in powers of chv. We will show that this is true. Let us represent the polynomial ( ) n R shv as follows That is, from polynomial 2 1 ( ) R ch v for odd degrees shv, we take out the accumulation factor shv , from the polynomial function 2 2 ( ) R ch v for even degrees shv, we take out the accumulation factor 2 .
ch v Integrating, we shall obtain: We consider the integral of function (6) near a singular point w i     .
We introduce a new independent variable  in the following way: Using formula (7), we rewrite the expression for chv in the following form: The expression for chv can be represented as the following factorization Components of perturbing acceleration X, Y, Z in the vicinity of a singular point w i     are represented by factorizations of the following form: where   B  -a function without negative degree  because after replacing the independent regularizing variable w by v , the components of the perturbing acceleration have the form: where chv is determined by factorizations (18). Perturbations in auxiliary quantities i C in a neighborhood of a singular point are defined as follows 0 4 5 6 ( ) ,  is a certain complex number.
The components of the perturbing acceleration X, Y, Z, taking into account expression (10), are representable in the form of factorizations for the neighborhood of points shv i   ; chv=0 in the form where ( ) R chv  -a polynomial in powers of chv. Let us represent dependence 4 5 6 , , Perturbations in auxiliary variables i C will have the following form 4 5 6 ( ) Substituting expressions (24) and (25)  Extending the above proof for a neighborhood of singular points to the general case, for any points of the complex plane v, we have that the coordinates that make up the regularized velocity and time in the second approximation for the trajectories of close passage from the disturbing body are representable by polynomials with respect to the new regularizing variable v.

Conclusions
The article is devoted to the determination of second-order perturbations in rectangular coordinates and components of the body motion to be under study. The main difficulty in solving this problem was the choice of a system of differential equations of perturbed motion, the coefficients of the projections of the perturbing acceleration are entire functions with respect to the independent regularizing variable. This circumstance allows constructing a unified algorithm for determining perturbations of the second and higher order in the form of finite polynomials with respect to some regularizing variables that are selected at each stage of approximation. The number of approximations is determined by the given accuracy. It is rigorously proven that the introduction of a new regularizing variable provides a representation of the right-hand sides of the system of differential equations of perturbed motion by finite polynomials. Special points are used to reduce the degree of approximating polynomials, as well as to choose regularizing variables.