Chapter Description Author(s)
The Tractrix and Similar Curves We generalize the classical tractrix problem (Gottfried Wilhelm Leibnitz, 17th century: "Given a watch attached to a chain, what is the orbit in the plane described by the watch as the endpoint of the chain is pulled along a straight line?") to compute the orbit of a toy pulled by a child, and then we compute the orbit of a dog which attacks a jogger. We use MATLAB for numerical solving two similar system of differential equations and show also how the motions can be visualized by MATLAB. W. Gander, S. Bartoň, J. Hřebíček
Trajectory of a Spinning Tennis Ball This chapter shows how to describe and visualize the variant of a motion of tennis ball in the air. Both Maple and MATLAB solutions are discussed. We start with the simplest model in the vacuum, then we modify it to a model of a tennis ball moving in the air and finally to the model of a spinning tennis ball in the air. We assume the conditions near earth surface. F.Klvaňa
The Illumination Problem Let us consider two lights on a horizontal road, given the heights of the lamps and their illumination powers. Suppose that we also know the distance between them and that there are no more lamps around. The goal is to find a point X between the two lamps which is minimally illuminated. We are also looking for the optimal heights of the lamps to have the best illumination on the whole road. The problem is solved using Maple. S. Bartoň, D. Gruntz
Orbits in the Planar Three-Body Problem The planar three-body problem is the problem of describing the motion of three point masses in the plane under their mutual Newtonian gravitation. It is a popular application of numerical integration of system of ordinary differential equations since most solutions are too complex to be described in terms of known functions. Maple and MATLAB can be used efficiently to construct and display numerical solutions of the planar three-body problem. First we will do straightforward numerical integration in MATLAB. Although for most initial conditions this approach will quickly produce an initial segment of the solution, it will usually fail at a sufficiently close encounter of two bodies, owing to the singularity at the corresponding collision. Therefore we use a set of variables that amounts to automatically regularizing each of the three types of close encounter whenever they occur. Owing to the complexity of the transformed equations of motion, the Hamiltonian formalism will be used for deriving these equations. Then Maple's capability of differentiating algorithms (automatic differentiation) will be used to generate the regularized equations of motion. D. Gruntz, J. Waldvogel
The Internal Field in Semiconductors Suppose we have a semiconductor of a given length, which is doped with a concentration of electrically active impurities (acceptor and donor). Let it be connected to a given external potential. The goal is to find the external potential inside the semiconductor as a function of the position between the ends. As a background we use Boltzmann's statistics for electrons and holes. The mathematical description leads to nonlinear Poisson equation. We show the solution using simple numerical algorithm implemented in MATLAB and even more simple solution in Maple using the new Maple capabilities for solving boundary value problems. F. Klvaňa, J. Pešl
Some Least Squares Problems This chapter considers some least squares problems, that arise in quality control in manufacturing using coordinate measurement techniques. In mass production machines are producing parts and it is important to know if the output satisfies the quality requirements. Therefore some sample parts are usually taken out of the production line, measured carefully and compared with the nominal features. If they do not fit the specified tolerances, the machine may have to be serviced and adjusted to produce better parts. We show how to compute several least squares fits -- especially for fitting lines, rectangles and squares in the plane and fitting hyperplanes in space. W. Gander, U. von Matt
The Generalized Billiard Problem Given a billiard table and two balls on it, from which direction should the first ball be struck, so that it rebounds off the rim of the table, and then impacts the second ball? This is a very simple mathematical problem for rectangle tables and has be solved also for circular tables and some other specific curves. In this chapter we solve that problem in general, for any parametrically described billiard table, using the generalized reflection method and the shortest trajectory method. The computation is done in Maple and if the curve is simple enough that the analytic solution exists, then the analytic solution is given; the numerical approximation is done only if it is really needed. S. Bartoň
Mirror Curves The generalized billiard problem brings another interesting mathematical problem: Given a starting point and applying the generalized reflection method, we will get a curve of mirrored points (function of the tangent point, where the ball hits the rim). This chapter studies these curves using Maple and then solves the inverse problem - to compute the starting points curve from the mirror curve and given tangent point. The solution can be obtained numerically using Maple, but for some cases also the analytical solution using geometrical tricks is shown. S. Bartoň
Smoothing Filters In many applications one is measuring a variable that is both slowly varying and corrupted by random noise. Then it is often desirable to apply a smoothing filter to the measured data in order to reconstruct the underlying smooth function. We assume that the noise is independent of the observed variable and that the noise obeys a normal distribution with zero mean and given variation. We discuss two possible approaches - the Savitzky-Golay filter and a least squares filter. W. Gander, U. von Matt
The Radar Problem The controlling system for a multiradar display, in an air traffic long-distance control center, receives different information from different kinds of radars on the globe. The information coming from each radar contains among others the coordinates of the airplane which is "seen" by the radar. We show how to reconstruct the most probable position in the "absolute" coordinate system. The transformations are computed using MATLAB. S. Bartoň, I. Daler
Conformal Mapping of a Circle Mapping techniques are mathematical methods which are frequently applied for solving fluid flow problems in the interior involving bodies of nonregular shape. Since the advent of supercomputers such techniques have become quite important in the context of numerical grid generation. In this chapter we shall demonstrate how the mathematical transformations required in applying mapping methods can be handled elegantly by means of a language for symbolic computation and computer algebra. Rather than choosing a large physical problem that would be beyond the scope of this book, we select a very simple application of comformal mapping to illustrate the essential steps involved. The problem is solved in MATLAB. H. J. Halin, L. Jaschke
The Spinning Top The motion of a spinning top - the well known childrens' toy - can be represented as a symmetric rigid rotor in a homogenous gravitational field. This allows us to set up a mathematical model, get the motion description in MATLAB and do the visualization. Many variants dependent of the input parameter combinations are shown. F.Klvaňa
The Calibration Problem When measuring gass pressure changes in a container, for example an engine cylinder or a gun, by means of a piezoelectric pressure transducer, highly relatively accurate values must be made available in order to obtain the specified absolute accuracy. For this, special measuring and calibrating techniques are necessary, which allow the quantitative determination of the dynamic measuring properties of the transducer. The output from the transducer is in electric voltage. Therefore we must perform the calibration of the transducer so as to finally get the pressure. This is not difficult when we are working with a static pressure. The essential problem with dynamic pressure approach is in the development of a physical model which allows a mathematical description of the hydraulic pressure pulses. We show the solution of this problem in Maple and it enables us to calibrate a dynamic pulse pressure transducer in absolute pressure units. J. Buchar, J. Hřebíček
Heat Flow Problems The heat flow problems are a very important part of thermodynamics. The solution of these problems influences many other technical problems. The most important equation describing heat flow rules, is the heat equation (Fourier equation). The difficulty of the solution depends on the difficulty of the boundary and initial conditions. We show the solution in Maple for both main types of heat flow problems - the steady state problems and the time-dependent problems. S. Bartoň, J. Hřebíček
Modeling Penetration Phenomena Penetration phenomena are of interest in numerous areas. They are often associated with the problem of nuclear waste containment and with the protection of spacecraft or satellites from debris and/or meteorite impact. Formally the penetration is defined as the entrance of a projectile into the target without completing the passage through the body. The penetration phenomenon can be characterized according to the impact angle, the geometry and material characteristics of the target and the projectile and the striking velocity. We limit our considerations to the normal incidence impact of a long rod on a semi-infinite target. This model corresponds for example to the situation in which a very thick armor is penetrated by a high kinetic energy projectile. The most efficient method for the solution of this problem is the numerical modeling by the finite element method. In this spirit, we will investigate some penetration models which are treated using Maple. J. Buchar, J. Hřebíček
Heat Capacity of System of Bose Particles In this chapter we will study a system of Bose particles with nonzero mass, in low temperature near absolute zero, when an interesting effect of superfluidity (or also superconductivity of electrons) appears. The behavior is shown in Maple using the generalization of Riemann Zeta function and Bose-Einstein integrals. F.Klvaňa
Free Metal Compression Compression is a widely used basis process in metal forming. If compression is performed using two plate platens, the lateral sufrace is distorted. This successive forming operation is called die forming, and it is necessary to predict the distortion in advance in order to provide enough space to fit the distorted body into a specific die. In the chapter we will restrcit our attention to metal rods with constant cross section, since nonconstant cross sections are not relevant in practice, and show the process in Maple. S. Bartoň
Gauss Quadrature In this chapter we will study how to compute Gauss quadrature rules with the help of Maple. The purpose is to approximate the integral of weighted function by a finite sum such that all polynomials to as high degree as possible are integrated correctly. We also consider the cases of Gauss-Radau and Gauss-Lobatto quadrature. The key role will play the orthogonal polynomials and Lanczos algorithm. U. von Matt
Symbolic Computation of Explicit Runge-Kutta Formulas In this chapter we show how Maple can be used to derive explicit Runge-Kutta formulas. Such formulas are used to solve systems of differential equations of first order. We show how the nonlinear system of equations for the coeffitients of the Runge-Kutta formulas are constructed and how such a system can be solved. We close this chapter by a overall procedure to construct Runge-Kutta formulas of a given size and order. We will see up to which size such a general purpose program is capable of solving the equations obtained. D. Gruntz
Transient Response of a Two-Phase Half-Wave Rectifier Electronic circuits are typically governed by linear differential equations with constant or time-dependent coeffitients. The numerical simulation of such system in the time-domain can be quite demanding, especially if the systems are very large and if they feature widely distributed eigenvalues. We would like to take an advantage of the analytical capabilities and accuracy of Maple i order to elegantly solve a small but tricky sample problem from the area of electronic circuits. It will be outlined why the problem is demanding in many ways. For the numerical solution by means of conventional programs a straightforward implementation of the mathematical model would not be sufficient. Instead some tricks will be used to overcome the several numerical difficulties to be discussed later on. This is why an unexperienced analyst most likely will not immediately succeed in performing this simulation study. H. J. Halin, R. Strebel
Circuits in Power Electronics Over the last few years high-power semiconductor devices with intristic turn-off capability have become available. These devices, called gate turn-off thyristors, consist of several layers of silicon with appropriate dotations. They are able to turn off currents of 1000 Amperes at thousands of volts within microseconds. For every fixed state of the thyristor switches the Kirchhoff's laws must be satisfied, and therefore the dynamical behavior of such a circuit is described by a system of linear ordinary differential equations with constant coeffitients, assuming linearity of the circuit elements. If the switches change their position the structure of the circuit changes, but the final state of the currents before switching determines the initial conditions after the switching. Therefore the mathematical model is a system of linear differential equations with piecewise constant coeffitients (if the switching time is neglected) and is solved here using MATLAB. J. Waldvogel
Newton's and Kepler's laws The goal of this chapter is to demonstrate the use of computer algebra in physics teaching. Seven practical examples from Newton's theory of gravity will be solved with support of Maple. All the examples will use no more than the famous Newton's law for gravitational force or te corresponding potential energy. We will show the problem of equilibrium of two and three forces, gravitation of the massive line segment or the solution for the earth satellite problem. S. Bartoň
Least Squares Fit of Point Clouds We solve a Procrustes Problem: We consider a least squares problem in coordinate metrology: nominal points of a workpiece are given by their exact coordinates from construction plans of a workpiece. Suppose now that a coordinate measuring machine gathers the same points of another workpiece in a different frame than the frame of reference. The problem we want to solve using MATLAB is to find a frame transformation which maps the given nominal points onto the measured points. W. Gander
Modeling Social Processes The general approach to modeling the evolution of a social system is based on an aspect space, which is defined as linear vector space. Its vectors characterize members of the social system with respect to their behavior in society or incorporationinto a social group. These aspects can be occupation, bias towards a certain political party, standard of living, level of income etc. Vectors of the aspect space characterize the incorporation of a member of the system into a certain group with respect to the given aspect. We introduce also the transient probability of the change in opinion of members represented by the given aspect. This leads to differential equation called master equation, which is the fundamental equation describing one class of social systems. We show in Maple how to use it to model processes with spatial structure (e.g. population migration). J. Hřebíček, T. Pitner
Contour Plots of Analytic Functions There are two easy ways in MATLAB to construct contour plots of analytic functions, i.e. lines of constant modulus and constant phase. One is to use the MATLAB contour command for functions of two variables, another to solve the differential equations satisfied by the contour lines. This is illustrated here for the partial sums of exponential series. In this case the lines of constant modulus are of interest in the numerical solution of ordinary differential equations, where they delineate regions of absolute stability for the Taylor expansion method of given order and also for explicit Runge-Kutta methods. W. Gautschi, J. Waldvogel
Non Linear Least Squares: Finding the most accurate location of an aircraft Consider a simplified typical situation of navigation in modern aircraft: The airplane is in an unknown position and receives signals from various beacons. Some beacons allow the airplane to read the angle from which the signal is coming. Other type of beacons allow to measure the distance from the airplane to the beacon. Each of the measures is given with an estimate of its error. The main purpose of this chapter is to develop a method for computing the most likely position of the aircraft based on the information available. We implement the solution in Maple and perform also the sensitivity analysis. G. H. Gonnet
Computing Plane Sundials There are many types of sundials. Virtually anything casting a shadow can be made into a sundial, but often not very accurate. The aim of this chapter is to convey the mathematics which is necessary to design accurate plane sundials. It is important to note, that to be accurate, the sundial must be specially designed for the spot it is to be used in and must also be pointed in the right direction. The algorithms written in MATLAB allow the reader to perform these calculations for his own sundial. M. Oettli, H. Schilt
Agriculture Kinematics In this chapter we show how to use Maple to simulate the movement and study the kinematics of a straw press feeder, used to sweep straw on the ground to the press. It has attached three scrapers of equal length. A chain moves around two cog-wheels (of the same radius) with the constant velocity. We would like to compute the position, the velocity and the acceleration of the ends of the scrapers as a function of time. The results will be plotted and also a simulation of the movements of the machine will be presented. S. Bartoň, Z. Hakl
The Catenary Curve In this chapter we derive and solve the differential equation for the catenary curve. Givne the end points and the length of the chain as boundary conditions we show how to compute a specific curve by solving the resulting nonlinear system in an elegant and machine independent foolproof way. We discuss the speed of convergence of the fixed-point iteration. Maple and MATLAB are used to support the solution process. W. Gander, U. Oswald
Least Squares Fit with Piecewise Functions We consider in this chapter the problem to fit piecewise polynomials to a data set with possibly different degrees and free knots. We also consider an example of fitting piecewise two exponential functions. We develop MATLAB programs to compute the fit in the least squares sense and demonstrate with some typical examples the sensitivity of the problem. W. Gander, S. Bartoň
Portfolio Problems - Solved Online Suppose we wish to invest some fixed capital into a portfolio. The question we would like to answer in this chapter is: how to partition the investment under the stocks to maximize the expected return and minimize the volatilities? This is a variant of the model published in Maple Applications Center, with gathering the data online using the Sockets package of Maple. The main purpose of this chapter is to show how to obtain and process on-line data from the web. We discuss the advantages and problems of this new approach. J. Hřebíček, J. Pešl