Elliptic Functions Lay the Foundations for Modern Physics

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Elliptic Functions Lay the Foundations for Modern Physics

Overview

Elliptic functions are considered a special class of analytic mathematical functions that are used to analyze and solve problems in physics, astronomy, chemistry, and engineering. More specifically, elliptic functions (known to modern mathematicians as elliptic integrals) are a large class of integrals related to, and containing among them, the expression for the arc of an ellipse. The advancement of elliptic functions during the nineteenth century provided mathematical precision in calculations required for discoveries in astronomy, physics, algebraic geometry, and topology. In addition to their use in applied mathematics, the development of the theory of elliptic functions also spurred the study of functions of complex variables and provided a bridge between pure and applied mathematics.

Background

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 descriptions 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 description 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 description and application of elliptical functions throughout his career and, in 1829, set forth a well-regarded description 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 descriptions and predictions of pendular movement.

Elliptical functions also allowed more accurate descriptions 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 descriptions 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.

K. LEE LERNER