Summary Poincaré made many contributions to different fields of pure and applied mathematics such as:
celestial mechanics,
fluid mechanics,
optics,
electricity,
telegraphy,
capillarity,
elasticity,
thermodynamics,
potential theory,
Quantum mechanics,
theory of relativity and
physical cosmology. Among the specific topics he contributed to are the following: •
algebraic topology (a field that Poincaré virtually invented) • the theory of analytic functions of
several complex variables •
the theory of abelian functions •
algebraic geometry • the
Poincaré conjecture, proven in 2003 by
Grigori Perelman. •
Poincaré recurrence theorem •
hyperbolic geometry •
Fuchsian groups •
number theory • the
three-body problem • the theory of
diophantine equations •
electromagnetism •
special relativity • the
fundamental group • In the field of
differential equations Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the
Poincaré sphere and the
Poincaré map. • Poincaré on "everybody's belief" in the
Normal Law of Errors (see
normal distribution for an account of that "law") • Published an influential paper providing a novel mathematical argument in support of
quantum mechanics.
Three-body problem The problem of finding the general solution to the motion of more than two orbiting bodies in the
Solar System had eluded mathematicians since
Newton's time. This was known originally as the three-body problem and later the
n-body problem, where
n is any number of more than two orbiting bodies. The
n-body solution was considered very important and challenging at the close of the 19th century. Indeed, in 1887, in honour of his 60th birthday,
Oscar II, King of Sweden, advised by
Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific: Given a system of arbitrarily many mass points that attract each according to
Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series
converges uniformly. In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished
Karl Weierstrass, said,
"This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics." (The first version of his contribution even contained a serious error; for details see the article by Diacu and the book by
Barrow-Green). The version finally printed contained many important ideas which led to the
theory of chaos. The problem as stated originally was finally solved by
Karl F. Sundman for
n = 3 in 1912 and was generalised to the case of
n > 3 bodies by
Qiudong Wang in the 1990s. The series solutions have very slow convergence. It would take millions of terms to determine the motion of the particles for even very short intervals of time, so they are unusable in numerical work. and introduced the hypothesis of
length contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (see
Michelson–Morley experiment). Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In
The Measure of Time (1898), Poincaré said, "A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the constancy of the speed of light as a
postulate to give physical theories the simplest form. Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.
Principle of relativity and Lorentz transformations In 1881 Poincaré described
hyperbolic geometry in terms of the
hyperboloid model, formulating transformations leaving invariant the
Lorentz interval x^2+y^2-z^2=-1, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions. In addition, Poincaré's other models of hyperbolic geometry (
Poincaré disk model,
Poincaré half-plane model) as well as the
Beltrami–Klein model can be related to the relativistic velocity space (see
Gyrovector space). In 1892 Poincaré developed a
mathematical theory of
light including
polarization. His vision of the action of polarizers and retarders, acting on a sphere representing polarized states, is called the
Poincaré sphere. It was shown that the Poincaré sphere possesses an underlying Lorentzian symmetry, by which it can be used as a geometrical representation of Lorentz transformations and velocity additions. He discussed the "principle of relative motion" in two papers in 1900 and named it the
principle of relativity in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest. In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance". In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz. In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all—it was necessary to make the Lorentz transformation form a group—and he gave what is now known as the relativistic velocity-addition law. Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote: The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form: ::x^\prime = k\ell\left(x + \varepsilon t\right)\!,\;t^\prime = k\ell\left(t + \varepsilon x\right)\!,\;y^\prime = \ell y,\;z^\prime = \ell z,\;k = 1/\sqrt{1-\varepsilon^2}. and showed that the arbitrary function \ell\left(\varepsilon\right) must be unity for all \varepsilon (Lorentz had set \ell = 1 by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination x^2+ y^2+ z^2- c^2t^2 is
invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct\sqrt{-1} as a fourth imaginary coordinate, and he used an early form of
four-vectors. Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit. So it was
Hermann Minkowski who worked out the consequences of this notion in 1907.
Mass–energy relation Like
others before, Poincaré (1900) discovered a relation between
mass and
electromagnetic energy. While studying the conflict between the
action/reaction principle and
Lorentz ether theory, he tried to determine whether the
center of gravity still moves with a uniform velocity when electromagnetic fields are included. the possibility that energy carries mass and criticized his own solution to compensate the above-mentioned problems: In the above quote he refers to the Hertz assumption of total aether entrainment that was falsified by the
Fizeau experiment but that experiment does indeed show that light is partially "carried along" with a substance. Finally in 1908 Poincaré's paradox, without using any compensating mechanism within the ether. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem, Einstein noted that Poincaré's formulation and his own from 1906 were mathematically equivalent.
Gravitational waves In 1905 Poincaré first proposed
gravitational waves (
ondes gravifiques) emanating from a body and propagating at the speed of light. He wrote:
Poincaré and Einstein Einstein's first paper on relativity was published three months after Poincaré's short paper, In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 titled "
Geometrie und Erfahrung (Geometry and Experience)" in connection with
non-Euclidean geometry, but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognized that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ....".
Assessments on Poincaré and relativity Poincaré's work in the development of special relativity is well recognised, Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks at rest in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time. While this is the view of most historians, a minority go much further, such as
E. T. Whittaker, who held that Poincaré and Lorentz were the true discoverers of relativity.
Algebra and number theory Poincaré introduced
group theory to physics, and was the first to study the group of
Lorentz transformations. He also made major contributions to the theory of discrete groups and their representations.
Topology The subject is clearly defined by
Felix Klein in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested by
Johann Benedict Listing, instead of previously used "Analysis situs". Some important concepts were introduced by
Enrico Betti and
Bernhard Riemann. But the foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894. His research in
geometry led to the abstract topological definition of
homotopy and
homology. He also first introduced the basic concepts and invariants of combinatorial topology, such as
Betti numbers and the
fundamental group. Poincaré proved a formula relating the number of edges,
vertices and faces of
n-dimensional
polyhedron (the
Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.
Astronomy and celestial mechanics Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of
algebraic and
transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since
Isaac Newton. These monographs include an idea of Poincaré, which later became the basis for mathematical "
chaos theory" (see, in particular, the
Poincaré recurrence theorem) and the general theory of
dynamical systems. Poincaré authored important works on
astronomy for the
equilibrium figures of a gravitating rotating fluid. He introduced the important concept of
bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).
Differential equations and mathematical physics After defending his doctoral thesis on the study of singular points of the system of
differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "
qualitative theory of differential equations". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (
saddle,
focus,
center,
node), introduced the concept of a
limit cycle and the
loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the
finite-difference equations, he created a new direction – the
asymptotic analysis of the solutions. He applied all these achievements to study practical problems of
mathematical physics and
celestial mechanics, and the methods used were the basis of its topological works. File: Phase Portrait Sadle.svg | Saddle File: Phase Portrait Stable Focus.svg | Focus File: Phase portrait center.svg | Center File: Phase Portrait Stable Node.svg | Node ==Character==