Linearization of the nonlinear equations and iterative solution is the most well-known scheme in many engineering problems. For geodetic applications of the LEO satellites, e.g. the Earth's gravity field recovery one needs to provide an initial guess of the satellite location or the so-called reference orbit. Numerical integration can be utilized to generate the reference orbit if a satellite's state vector, i.e. position and velocity is known at a reference epoch. However the numerically integrated orbit deviated from the real orbit due to imperfect force models. The more accurate the reference orbit the less linearization error occurs. The deviation between the reference and real orbit can be minimized using the least squares method. Different analytical and numerical techniques have been developed for calculation of the design matrix of the least squares method. Herein we have generalized the idea of the Lagrange Coefficients for determination of the design matrix's entries in the gravitational field of an attracting inhomogeneous mass body. Numerical implementation of the proposed method shows its high performance.

In this paper we demonstrate the existence of a set of polynomials Pi, 1£ i £ n , which are positive semi-definite on an interval [a, b] and satisfy, partially, the conditions of polynomials found in the Lagrange interpolation process. In other words, if a=a1<…<an=b is a given finite sequence of real numbers, then Pi (aj)=dij (dij is the Kronecker delta symbol); moreover, the sum of Pi 's is identically 1.

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler–Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical Euler–Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.

The conflict between the Newtonian and the Leibnizian picture of spacetime is well-known and widely discussed in the literature. However, two competing metaphysical pictures of spacetime have not been addressed so far: The Newtonian and the Lagrangian pictures. According to the Newtonian picture, spacetime can be considered as the time evolution of a spatial hypersurface. According to the Lagrangian picture, spacetime is a unified whole that can be examined in an all-at-once manner, and its (temporal) evolution is not meaningful, except locally or quasi-locally. Although the Newtonian picture has been the dominant metaphysical picture since Newton's time, our explicit familiarity with these two competing pictures is recent. It is rooted in Smolin's criticism of modern cosmology and Wharton's research on the foundations of quantum mechanics. Here, we describe the views of Smolin and Wharton, then we will argue that two famous formulations of general relativity, namely the Hamiltonian formulation and the Lagrangian formulation, require the Newtonian and the Lagrangian picture, respectively. Also, we will argue that: (1) the Lagrangian picture is more compatible with the philosophy of relativity theory than the Newtonian picture, while the Newtonian picture is only acceptable for practical and computational purposes. And, (2) the Lagrangian picture is universal and applicable to any spacetime, while the Newtonian picture can only be used for certain spacetimes. Therefore, the Lagrangian picture is a more plausible metaphysical picture than the Newtonian.

We elaborate a unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated conditions when the fundamental geometric objects (anchor, metric and linear connection, almost symplectic, and related almost complex structures) may be canonically defined by an N-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructions are modelled on nonholonomic manifolds provided with nonintegrable distributions and related chains of exact sequences of submanifolds defining N-connections. We investigate the main properties of the Lagrange, Hamilton, Finsler-Riemann and Einstein-Cartan algebroids, construct and analyze exact solutions describing such objects.

In this work, a new type of interpolation formula is introduced. These formulas can be an extension of the Lagrange interpolation formula. The error of this new type of interpolation is calculated. In order to display efficiency of the proposed formulas, three numerical examples are presented.

In this paper, perturbation technique and Lagrange expansion to solve the nonlinear algebriac equations is introduced. Also Some examples are provided to illustrate the simplicity and efficiency of the new proposed methods.