SOME ELLIPTIC FUNCTIONS HISTORY<Although not called elliptical functions until the nineteenth century, modern study of elliptical functions began in the middle of the seventeenth century. Several mathematicians published works examining the arc length of an elliptical path and Sir Isaac Newton (1642-1727) published works regarding the mathematics of elliptical orbits. As a particular consequence of Newton's work, the mathematical development of the elliptic functions and integrals emerged from centuries of struggle to accurately (i.e., mathematically) explain the mechanics of motion, including the motions of the Sun and planets.

In addition to descrïptions of the elliptical paths and orbits of moving bodies, in 1679 Swiss mathematician Johann Bernoulli (1667-1748), while attempting to mathematically describe a spiral path, found that the true descrïption for the deformation of an compressed elastic rod was elliptical. Bernoulli determined that the resulting curve describing the deformation was, by definition, what would later be known as an elliptical function.

During the later years of the eighteenth century, French mathematician and astronomer Pierre Simon Marquis de Laplace (1749-1827), the subsequent discoverer of the Laplace theorem that bears his name, and others studied unexplained variations in the orbits of Jupiter and Saturn. These enigmatic changes (a contraction in Jupiter's orbit and an expansion in Saturn's orbit) seemed an important discrepancy to the Newtonian-based cosmological models of eighteenth- and nineteenth-century science and philosophy that assumed a static and unchanging universe. Although Laplace eventually succeeded in accounting for the orbital variations with a periodicity predicted by Newtonian gravity (and hence compatible with existing cosmological models), his work stirred a greater scrutiny of celestial mechanics and set the stage for the advancement of elliptical functions as mechanisms to calculate and predict celestial movements.

Laplace also published works that dealt with the precise motion of the Moon around Earth, a problem that had frustrated earlier generations of mathematicians. Laplace was the first to account for the influence of the Sun on the lunar orbit (i.e., the influence of the Sun's gravity on the two body Earth-Moon system) and of the Sun and planets on ocean tides . Laplace's work, made popular with his 1796 publication of Exposition du Système du Monde, stirred interest in developing refined mathematical tools that would enable astronomers to more easily and fully study perturbations of celestial movement.

This heightened interest set the stage for the ideas and terminology embodied by elliptical functions that were to evolve from Swiss mathematician Leonhard Euler's (1707-1783) elegant explanations of mechanics using differential equations, German Johann Carl Friedrich Gauss's (1777-1855) work that anticipated many properties of elliptical functions, and of the Italian-born Joseph-Louis Lagrange's (1736-1813) theories of functions related to celestial mechanics.

Impact

In the nineteenth century early understanding and application of elliptic functions was principally nourished by the work of the great Norwegian mathematician Niels Henrich Abel (1802-1829) and Prussian mathematician Karl Gustav Jacob Jacobi (1804-1851). Although Abel and Jacobi were both pioneers of modern mathematics, they came from vastly different circumstances, drawn together in history only through their competitive quest to understand and describe the properties of elliptical functions.

Abel's brief but brilliant career was on constant battle with destitution. When Abel was 18, the death of his father put Abel's educational prospects in doubt. With help from professors who recognized his mathematical talent, Abel entered the University of Oslo in 1821. By 1923, despite his modest circumstances, Abel paid for the publication of solutions to mathematical problems that had previously vexed mathematicians for hundreds of years. These solutions also promoted the work of others, including Jacobi's development of integral equations. Despite his brilliance, Abel wandered through European academia, unable to find a permanent position. He lived hand-to-mouth by tutoring and substitute teaching. Eventually, while working in Berlin, Abel began to dedicate his energies to the study of elliptical functions.

After reading Jacobi's publications regarding transformations of elliptic integrals, Abel realized that Jacobi's work was based, in part, on Abel's own unpublished insights, Able scrambled to compete with Jacobi and managed to swiftly publish several papers on elliptical functions, including a 1827 work titled Recherches sur les fonctions elliptiques. Abel's discovery that elliptical functions were actually the inverse of elliptical integrals brought him world-wide acclaim and fame. Before he could reap the rewards of his success, however, and just as he was about to be appointed to a professorship, Abel succumbed to tuberculosis contracted during his travels.

In contrast, Jacobi (1804-1851) enjoyed a stable academic career. First at the University of Königsberg and then, 15 years after the tragic death of Abel, in Berlin. Jacobi worked on the descrïption and application of elliptical functions throughout his career and, in 1829, set forth a well-regarded descrïption of the functions in his Fundamenta Nova Theoria Functionum Ellipticarum.

In 1830, Paris Academy awarded the Academy's Grand Prix to Abel (posthumously) and Jacobi for their outstanding work.

French mathematician Adrien Legendre (1752-1833) also made substantial contributions to the study of elliptical integrals. In fact, the term elliptic function arguably first appeared in Lengendre's 1811 publication Exercises du Calcul Intégral.

Although elliptic functions were simple in form, defined as r(x, p(x))dx where r(x,y) is a rational function in two variables and p(x) is a 3rd or 4th degree polynomial without repeated roots, the development of elliptical functions had profound consequences on the analysis of the mechanics of motion. One problem, for example, that had confounded engineers and scientists was the ability to accurately and quickly determine the perimeter of an elliptical oval or the swing of a pendulum. Because of non-linear elements in these problems, the problems could not be easily solved using standard elementary functions, and application of elliptic functions as an analytical tool was required in order to make accurate descrïptions and predictions of pendular movement.

Elliptical functions also allowed more accurate descrïptions and predictions of the celestial mechanics of Keplerian orbital motions (i.e., gravitationally bound two-body systems or the bound motion of a one body system interacting with an attractive central force). The Sun and planets comprise such a two-body system that can be described by elliptical functions and, in accord with Kepler's laws of planetary motion, the application of elliptic functions provided very precise descrïptions of planetary motions that allowed the subsequent calculation of perturbations in those orbits caused by other bodies in the solar system.

The great power of elliptical functions was their ability to elegantly and accurately describe rotational dynamics, in particular, the highly complicated motions of celestial bodies that consumed nineteenth-century astronomers. Accordingly, the theory of perturbations and the understanding of planetary orbits owed much to the development of elliptic functions because they allowed astronomers to plot graphs of the orbits and distances traveled by a planet or comet along an orbital path as a function of time. More than a century after their articulation, elliptic integrals are still used to calculate spacecraft trajectories—especially those sent out on interplanetary missions.

In the middle of the nineteenth century, irregularities in the orbit of Uranus prompted astronomers and mathematicians to seek the cause of such perturbations. Precise calculations of Uranus' orbit using elliptical integrals showed that the perturbations could be explained by the presence of an undiscovered planet. Although some of these calculations were completed as early as 1845 by the British astronomer John Couch Adams (1819-1892) of Cambridge University, credit for the discovery of the planet, eventually named Neptune, was given to the brilliant French mathematician and astronomer Urbain Jean Joseph Le Verrier (1811-1877). In addition, Le Verrier's use of elliptical functions to describe a discrepancy in the orbital motion of Mercury (e.g., the advance of the perihelion of Mercury) became an important stimulus to the subsequent formation and proof of Albert Einstein's general theory of relativity.

French mathematician Jules Henri Poincaré's (1854-1912) use and advancement of elliptical functions in the fields of celestial mechanics (three-body problems) and in the emerging theories of light and electromagnetic waves provided significant contributions to the further development of Scottish physicist James Clerk Maxwell's (1831-1879) profoundly important equations of the electromagnetic field. Maxwell's equations describing the propagation of electromagnetic waves were derived from both hyperbolic and elliptical components (i.e., the unified equations make an important synthesis of hyperbolic and elliptic functions). Maxwell's equations became the essential key to understanding the scope of the electromagnetic spectrum and laid the essential foundations for the formation of twentieth-century quantum and relativity theory.

Beside wide-ranging use in number theory and celestial mechanics, elliptical functions continue to be used in many engineering applications and solving many problems in electromagnetism and gravitation.

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The terminology for elliptic integrals and functions has changed during their investigation. What were originally called elliptic functions are now called elliptic integrals and the term elliptic functions reserved for a different idea. We will therefore use modern terminology throughout this article to avoid confusion.

It is important to understand how mathematicians thought differently at different periods. Early algebraists had to prove their formulas by geometry. Similarly early workers with integration considered their problems solved if they could relate an integral to a geometric object.

Many integrals arose from attempts to solve mechanical problems. For example the period of a simple pendulum was found to be related to an integral which expressed arc length but no form could be found in terms of 'simple' functions. The same was true for the deflection of a thin elastic bar.

The study of elliptical integrals can be said to start in 1655 when Wallis began to study the arc length of an ellipse. In fact he considered the arc lengths of various cycloids and related these arc lengths to that of the ellipse. Both Wallis and Newton published an infinite series expansion for the arc length of the ellipse.

http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Elliptic_functions.html




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Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential—to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.....



http://aleph0.clarku.edu/~djoyce/hilbert/problems.html



very very small tic  

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