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Dimulai oleh The Houw Liong, November 30, 2017, 10:03:01 AM

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The Houw Liong


The Houw Liong

Henri Poincaré

[Picture of Poincare] (April 29, 1854 - July 17, 1912)
Le savant n'étudie pas la nature parce que cela est utile; il l'étudie parce qu'il y prend plaisir et il y prend plaisir parce qu'elle est belle. Si la nature n'était pas belle, elle ne vaudrait pas la peine d'être connue, la vie ne vaudrait pas la peine d'être vécue.
   - original French version of the quote on the main page
        taken from Science et méthode
Jules Henri Poincaré was dubbed by E. T. Bell as the "Last Universalist", a man who is at ease in all branches of mathematics, both pure and applied. Poincaré was one of these rare savants who was able to make many major contributions to such diverse fields as analysis, algebra, topology, astronomy, and theoretical physics. Like Gauss, another universalist, his mind was constantly brimming with highly creative ideas, but unlike Gauss, he published extensively. Among his many published works are several highly readable popular pieces that tried to give the general public a flavor of the workings of science.
The ideas of dynamical chaos was first glimpsed by Poincaré when he entered a contest sponsored by the king of Sweden. One of the questions in this contest was to show rigorously that the solar system as modeled by Newton's equations is dynamically stable. The question was nothing more than a generalization of the famous three body problem, which was considered one of the most difficult problems in mathematical physics. In essence, the three body problem consists of nine simultaneous differential equations. The difficulty was in showing that a solution in terms of invariants converges. While Poincaré did not succeed in giving a complete solution, his work was so impressive that he was awarded the prize anyway. The distinguished Weierstrass, who was one of the judges, 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." A lively account of this event is given in Newton's Clock: Chaos in the Solar System.

To show how visionary Poincaré was, it is perhaps best if he described the Hallmark of Chaos - sensitive dependence on initial conditions - in his own words:

If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. - in a 1903 essay "Science and Method"
A biography that gives more information about his other mathematical achievements besides chaos can be found in this nice History of Mathematics site under Poincaré.

(thanks to Etienne Forest for corrections)


Bell, E. T., Men of Mathematics, Simon & Schuster, 1986.

Peterson, Ivars, Newton's Clock: Chaos in the Solar System, W. H. Freeman, 1993.

Poincaré, Henri, The Value of Science, Dover, 1958.
Science and Hypothesis, Dover, 1952.
New Methods of Celestial Mechanics, AIP, 1993.

The Houw Liong

Poincaré is best known for his critique of logicism and for his geometric conventionalism. The two traditions interpreting Poincaré's work thus reflect, on the one hand, a philosophy of mathematics that endorses his intuitionist tendency and his polemics against logicism or formalism, and, on the other hand, his conventionalism both in the philosophy of science and in a broad linguistic sense. In reality, these intuitionistic and formal aspects are two sides of the same coin, given that Poincaré always supports a single position aimed at a reconstruction of the process of understanding scientific theories (see Heinzmann 2010). Neither formalistic nor intuitionist nor empiricist, Poincare opens a path lying between a realist and an anti-realist theory of mathematical knowledge.


The Houw Liong

Poincaré, celestial mechanics, dynamical-systems theory and "chaos"☆
Author links open overlay panelPhilipHolmes
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As demonstrated by the success of James Gleick's recent book [1987], there is considerable interest in the scientific community and among the general public in "chaos" and the "new science" which is supposed to accompany it. However, as usual, it is not easy to separate hyperbole from fact. In an attempt to do this, I will offer a precise definition of chaos in the context of differential equations: mathematical models which, since Newton, have played a vital role in scientific discovery. I will show how the classical problems of celestial mechanics led Poincaré to ask fundamental questions on the qualitative behavior of differential equations, and to realize that chaotic orbits would provide obstructions to the conventional methods of solving them.

In a major paper which appeared almost exactly one hundred years ago, Poincaré studied mechanical systems with two degrees of freedom and identified an important class of solutions, now called transverse homoclinic orbits, the existence of which implies the system has no analytic integrals of motion other than the total (Hamiltonian) energy. I will explain these terms and outline the history of subsequent developments of these ideas by Birkhoff, Cartwright, Littlewood, Levinson and Smale, and describe how the ideas of Melnikov have made possible an "analytical algorithm" for the detection of chaos and proof of nonintegrability in wide classes of perturbed Hamiltonian systems. I will discuss the physical implications of the mathematical statements that these methods afford. In the process, I will point out that, while there is a precise vocabulary and grammar of chaos, developed largely by mathematicians and steaming from Poincaré's work, it is not always easy to use it in speaking of the real world.


The Houw Liong