Analytic Inversion of Closed form Solutions of the Satellite’s J2 Problem

This report provides some closed form solutions -and their inversion- to a satellite’s bounded motion on the equatorial plane of a spheroidal attractor (planet) considering the J2 spherical zonal harmonic. The equatorial track of satellite motion- assuming the co-latitude φ fixed at π/2- is investigated: the relevant time laws and trajectories are evaluated as combinations of elliptic integrals of first, second, third kind and Jacobi elliptic functions. The new feature of this report is: from the inverse t = t(c) we get the period T of some functions c(t) of mechanical interest and then we construct the relevant c(t) expansion in Fourier series, in such a way performing the inversion. Such approach-which led to new formulations for time laws of a J2 problem- is benchmarked by applying it to the basic case of keplerian motion, finding again the classic results through our different analytic path.

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Closed-Form Solutions and Conserved Vectors of a Generalized (3+1)-Dimensional Breaking Soliton Equation of Engineering and Nonlinear Science

Mathematics ◽

10.3390/math8101692 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1692

Author(s):

Chaudry Masood Khalique ◽

Oke Davies Adeyemo

Keyword(s):

Analytical Solution ◽

Closed Form ◽

Elliptic Functions ◽

Lie Symmetry ◽

Jacobi Elliptic Functions ◽

Direct Integration ◽

Hyperbolic Functions ◽

Soliton Equation ◽

Closed Form Solutions ◽

Expansion Technique

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the (G′/G)-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov’s theorem.

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Symmetry transformation matrices for structures

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.1838 ◽

2007 ◽

Vol 463 (2082) ◽

pp. 1563-1583 ◽

Cited By ~ 3

Author(s):

Markus Pagitz ◽

Jonathan James

Keyword(s):

Fourier Series ◽

Closed Form ◽

Symmetry Transformation ◽

Symmetry Groups ◽

Vector Spherical Harmonics ◽

Alternative Derivation ◽

Closed Form Solutions ◽

Transformation Matrices ◽

Displacement Vectors ◽

Stiffness Matrices

Many structures in nature and engineering are symmetric. Depending on the degree of symmetry, it is possible to simplify the computations considerably by block diagonalizing the stiffness matrices. Closed-form solutions of transformation matrices for such block diagonalizations can be derived using group theory for arbitrary symmetry groups. This paper presents closed-form solutions of transformation matrices based on an alternative derivation. It is shown that transformation matrices for C nv and D nh groups can be obtained from a finite Fourier series decomposition of load and displacement vectors. Furthermore, it is shown that structures with tetrahedral, octahedral and icosahedral symmetries can be block diagonalized in an elegant way using vector spherical harmonics.

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A closed-form solution for Reissner planar finite-strain beam using Jacobi elliptic functions

International Journal of Solids and Structures ◽

10.1016/j.ijsolstr.2016.02.020 ◽

2016 ◽

Vol 87 ◽

pp. 153-166 ◽

Cited By ~ 14

Author(s):

Milan Batista

Keyword(s):

Closed Form ◽

Finite Strain ◽

Closed Form Solution ◽

Elliptic Functions ◽

Form Solution ◽

Jacobi Elliptic Functions

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Fourier Series Approximations toJ2-Bounded Equatorial Orbits

Mathematical Problems in Engineering ◽

10.1155/2014/568318 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Wei Wang ◽

Jianping Yuan ◽

Yanbin Zhao ◽

Zheng Chen ◽

Changchun Chen

Keyword(s):

Fourier Series ◽

Initial Conditions ◽

Polar Angle ◽

Elliptic Functions ◽

Dynamical Analysis ◽

Fourier Series Expansion ◽

Elementary Functions ◽

Mathematical Functions ◽

Jacobian Elliptic Functions ◽

Closed Form Solutions

The current paper offers a comprehensive dynamical analysis and Fourier series approximations ofJ2-bounded equatorial orbits. The initial conditions of heterogeneous families ofJ2-perturbed equatorial orbits are determined first. Then the characteristics of two types ofJ2-bounded orbits, namely, pseudo-elliptic orbit and critical circular orbit, are studied. Due to the ambiguity of the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions, showing little physical insight into the problem, a new scheme, termed Fourier series expansion, is adopted for approximation herein. Based on least-squares fitting to the coefficients, the solutions are expressed with arbitrary high-order Fourier series, since the radius and the flight time vary periodically as a function of the polar angle. As a consequence, the solutions can be written in terms of elementary functions such as cosines, rather than complex mathematical functions. Simulations enhance the proposed approximation method, showing bounded and negligible deviations. The approximation results show a promising prospect in preliminary orbits design, determination, and transfers for low-altitude spacecrafts.

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Closed form integration of the rotating plane pendulum nonlinear equation

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.34.2003.235 ◽

2003 ◽

Vol 34 (4) ◽

pp. 327-350 ◽

Cited By ~ 1

Author(s):

Giovanni Mingari Scarpello ◽

Daniele Ritelli

Keyword(s):

Differential Equation ◽

Nonlinear Equation ◽

Closed Form ◽

Nonlinear Differential Equation ◽

Vertical Plane ◽

Elliptic Functions ◽

Angular Speed ◽

Jacobi Elliptic Functions

The article deals with the nonlinear differential equation of the frictionless motion of a heavy pendulum swinging in a vertical plane which rotates at a fixed angular speed. The authors focused on its closed form integration by means of the Jacobi elliptic functions. This research took its origin by an autonomous work of the authors; this subject was also developed by [3], who did a treatment by far different from ours.

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Closed-form solution for column buckling optimization

10.21203/rs.3.rs-857140/v1 ◽

2021 ◽

Author(s):

Vladimir Kobelev

Keyword(s):

Closed Form ◽

Dimensional Analysis ◽

Optimization Problem ◽

Closed Form Solution ◽

Elliptic Functions ◽

Isoperimetric Inequalities ◽

Form Solution ◽

Algebraic Equations ◽

Closed Form Solutions ◽

Axial Forces

Abstract An optimization problem for a column, loaded by axial forces, whose direction and value remain constant, is studied in this article. The dimensional analysis introduces the dimensionless mass and rigidity factors, which simplicities the mathematical technique for the optimization problem. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier is superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental function. The principal results are the closed form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The proof of isoperimetric inequalities exploits the variational method and the Hölder inequality. The isoperimetric inequalities for Euler’s column are rigorously verified.

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