Symbolic-numerical methods of studying equilibrium positions of a gyrostat satellite

Methods of computer algebra are used to study properties of a nonlinear algebraic system that determines equilibrium positions of a gyrostat satellite moving along a circular orbit. Bifurcation values of parameters such that the number of equilibrium positions changes are found numerically. Detailed numerical analyses of evolution of existence domains of different numbers of equilibria in the space of dimensionless parameters are performed.     

Steady motions of gyrostat satellites and their stability

The steady motions of a rigid body rotating about its maximum principal axis of inertia, while the radius vector lies in the direction of its minimum principal axis of inertia, is known to be stable in the sense of Lyapunov. Due in part to their stowed configuration in launch vehicles, however, satellites typically have an initial rotation about their minimum principal axis of inertia. Such rotation may be unstable in the presence of some dissipations. This paper investigates the effect of momentum wheels on the stability of steady motions. It is proved that the momentum wheels increase the effective moment of inertia of the gyrostat-satellite system about some desired axis. Stability of the steady rotation about the desired axis can be established only for the case when the moment of inertia of the axis aligned with the radius vector is smaller than that of the axis of linear momentum. A new set of stability criteria is obtained which includes the effects of the coupling between the orbital and attitude dynamics and may be useful in the design of attitude control systems for large spacecraft in low Earth orbit     

Discretization of quasilinear evolution equations by computer algebra methods

An algorithmic approach to construction of finite difference schemes on regular grids developed by the first two authors is applied to quasilinear evolution equations in one spatial variable. The approach combines the finite volumes method, numerical integration, and difference elimination, which is done by means of computer algebra. As a concrete example, a difference scheme for the Korteweg-de Vries equation is constructed. This scheme is strongly consistent and absolutely stable. The numerical behavior of the scheme obtained is illustrated by solving a Cauchy problem.     

On a computer algebra technology

Modern computer algebra systems provide means for exact experimental calculations (including manipulations of formulas) that enable one to obtain a solution of a problem under examination for certain not very large initial data. The results can provide a basis for a conjecture concerning the general solution of the problem. A computer algebra system can also be helpful in verifying the conjecture.     

Calculation of first integrals of a two-dimensional ODE system near a degenerate stationary point by computer algebra tools

An autonomous system of ordinary differential equations resolved with respect to derivatives is considered. To study its local integrability in a neighborhood of a degenerate stationary point, an approach based on the power geometry method is used. In a previous work, for some two-dimensional system depending on five parameters, a complete set of conditions on these parameters that are necessary conditions of local integrability of the considered system near a strongly degenerate stationary point was found. In this work, it is shown that these conditions are sufficient for global integrability of the system, and the corresponding first integrals of motion are found by using computer algebra tools.     

On Melnikov's method in the study of chaotic motions of gyrostat

The homoclinic solutions of the attitude motion of a gyrostat with wheels along its principal axes are formulated using the procedure developed by Wittenburg. The generic Melnikov function for the attitude motion of a gyrostat subject to small non-linear damping torques and small periodic torques is derived. The derivation is based on the chaotic theory for a 'one and half' degrees of freedom system established by Wiggins and Shaw. The conditions for the physical parameters at the onset of the homoclinic solutions are given in detail. The chaotic criteria are investigated using Melnikov's function. In particular, the conditions of the physical parameters (moments of inertia, damping coefficients, amplitudes and frequencies of external excitation, moments of momentum of wheels) that ensure the convergence of Melnikov's integral are also worked out. In order to acquire some knowledge of long-term behaviours of the chaotic attitude motion, the fourth-order Runge-Kutta numerical algorithms are utilized to simulate its dynamics. The numerical experiment shows that the chaotic motions of the gyrostat are bounded, non-periodic and sensitive to initial conditions. One of the practical mechanical models of the gyrostat is the artificial satellite. The attitude motion of the three-axis stabilized gyrostat satellite, whose torque-free motion is the homoclinic orbits, will oscillate chaotically when it is subjected to the appropriate external disturbances.     

An investigation of data entry methods with a personal computer

The features of three methods of data entry were investigated and a trial of them was carried out using operators with differing amounts of keyboard experience. The first method was a simple system of character entry using a word processing package, and the second and third systems were written in a commercial data base language; one was designed to possess limited intelligence, the other moderate intelligence. The amount of time and computing expertise required to set up each method increased in parallel with its sophistication. The most sophisticated method offered theoretical advantages of reduced error rates and increased keying rates but these were not realized in practice. The limited intelligence method, in which data were entered into a screen image of the record form, was fastest overall and was most popular with all users. This method together with verification by dual keying will provide a convenient, rapid, and high fidelity method of data entry. The simplest method was found to be adequate for short forms but incurred high error rates with longer forms.     

Hamiltonian normalization in the restricted many-body problem by computer algebra methods

A symbolic algorithm for construction of a real canonical transformation that reduces the Hamiltonian determining motion of an autonomous two-degree-of-freedom system in a neighborhood of an equilibrium state to the normal form is discussed. The application of the algorithm to the restricted planar circular three-body problem is demonstrated. The expressions obtained for the coefficients of the Hamiltonian normal form substantiate results derived earlier by A. Deprit. Symbolic computations are performed in the computer algebra system Mathematica.     

Learning physics with a computer algebra system

Abstract   To become proficient problem-solvers, physics students need to form a coherent and flexible understanding of problem situations with which they are confronted. Still, many students have only a limited representation of the problems on which they are working. Therefore, an instructional approach was devised to promote students' understanding of these problems and to support them in forming associations between problem features and solution methods. The approach was based on using the computer algebra software Mathematica as a tool for problem solving and visualisation. An electrostatics course module was implemented based on this instructional approach, and this module was compared with a usual paper-and-pencil based one. Learning outcomes for both courses were not significantly different. The experimental course was found to impose a high cognitive load on the students. Based on the outcomes, proposals are made for ways in which the course could be improved.     

gTybalt—a free computer algebra system

This article documents the free computer algebra system “gTybalt”. The program is build on top of other packages, among others GiNaC, TeXmacs and Root. It offers the possibility of interactive symbolic calculations within the C++ programming language. Mathematical formulae are visualized using fonts.

Program summary

Title of program: gTybaltVersion: 1.0.0Catalogue identifier: ADSIProgram summary URL:http://cpc.cs.qub.ac.uk.summaries/ADSIProgram obtained from: CPC Program Library, Queen's University of Belfast, N. IrelandLicense: GNU Public LicenseComputers: allOperating system: GNU/LinuxProgram language: C++Distribution format: tar gzip fileNo. of bytes in distributed program, including test data, etc.: 499583Memory required to execute: 64 MB recommendedOther programs called: see Appendix AExternal files needed: noneKeywords: Symbolic calculations, Computer algebraNature of the physical problem: Symbolic calculations occur nowadays in all areas of science. gTybalt is a free computer algebra system based on the C++ language.Method of solution: gTybalt is a “bazaar”-style program, it relies on existing, freely-available packages for specific sub-tasks.Restrictions on complexity of the problem: gTybalt does not try to cover every domain of mathematics. Some desirable algorithms, like symbolic integration are not implemented. It can however easily be extended in new directions. Apart from that, standard restrictions due to the available hardware apply.Typical running time: Depending on the complexity of the problem.     

Application of computer algebra for the reconstruction of surfaces from their photometric stereo images

The problem of reconstructing a Lambertian surface from its two photometric stereo images is discussed. Previously, the solution to this problem was only obtained for a special choice of two light source directions. In this paper, using the computer algebra system Mathematica, the necessary and sufficient conditions for the unique reconstruction of the surface from its two images is analyzed in a more general setting. Photometric images of various surfaces are simulated, and the validity of the theoretical results is demonstrated.     

Methods for justifying arithmetic hypotheses and computer algebra

Computer algebra methods for justifying hypotheses in arithmetic geometry are considered. The specific feature of these methods is that they are designed for parametric problems. The methods developed can be used for solving computational problems and justifying hypotheses on equidistribution, as well as for the theory of algebraic curves over finite fields.     

星载计算机可重配平台设计

对Tcl／Tk语言的代码可重用特性在可重配置平台上的应用进行研究，讨论星载计算机可重配置特性的机理，并详细介绍这种面向用户的脱离底层代码操作的重配置平台的搭建步骤和代码实现，并给出该方案在一款星载数据处理系统可重配置设计上的应用结果．     

Cadabra: a field-theory motivated symbolic computer algebra system

Field theory is an area in physics with a deceptively compact notation. Although general purpose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions, this often leads to cumbersome input formats, unexpected side-effects, or the need for a lot of special-purpose code. This makes a direct translation of problems from paper to computer and back needlessly time-consuming and error-prone. A prototype computer algebra system is presented which features -like input, graph data structures, lists with Young-tableaux symmetries and a multiple-inheritance property system. The usefulness of this approach is illustrated with a number of explicit field-theory problems.     

Experimental investigation of snakelike motions of a three-link mechanism

A three-link joint mechanism is considered, which is able to move on a flat rough surface by changing its configuration. A model for this mechanism is presented, and its control system is described. The results of experiments with the three-link robot for various motion modes are presented.     

Low-dimensional modelling of dynamics via computer algebra

I describe a method, particularly suitable to implementation by computer algebra, for the derivation of low-dimensional models of dynamical systems. The method is systematic and is based upon centre manifold theory. Computer code for the algorithm is relatively simple, robust and flexible. The method is applied to two examples: one a straightforward pitchwork bifurcation, and one being the dynamics of thin fluid films.     

Construction of nonlocal linear vibration models using a computer algebra system

In the paper, on the examples of a string and beam, resonance vibrations in linear elastic systems that result from nonlocal boundary conditions (even if external forces are lacking) are considered. Formulas of exact solutions of the corresponding boundary-value problems are presented. These formulas are used for evaluation of the solutions and for construction of plots. Computer algebra system Mathematica was used for derivation and analysis of the exact solutions and for development of software packages designed for evaluation of the derived formulas and solution visualization. The authors believe that results of this study can be used to explain the causes for the vibration of the Tacoma and Volgograd bridges in 1940 and 2010, respectively.