# math4b-18 - Math 4B Lecture 18 June 3 2013 Doug Moore It is...

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Math 4B Lecture 18 June 3, 2013 Doug Moore

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It is a fact of life that many applications require the use of non- linear systems of differential equations. Only the simplest such systems can be solved explicitly in terms of elementary functions. More complicated nonlinear systems can often be solved numer- ically. The numerical solutions need to be supplemented by a study of the qualitative behaviour of the solutions. An important example is provided by the question of stability of the solar system: Will the planets stay in the neighborhood of the sun during the next several billion years (until the sun runs out of nuclear fuel and becomes a red giant), or will the planets gradually drift away? According to classical Newtonian mechanics, this problem can be formulated in terms of a system of nonlinear ordinary differential equations.
Under the assumption that there is only one planet and the sun, the dynamical system can be solved exactly: the planet must move around the sun in an ellipse, with the sun at one focus. However, if one tries to account for the gravitational pulls of all nine planets, the dynamical system becomes too complicated to solve exactly. The system of differential equations can be solved numerically to yield approximate values of the positions of the planets for the next several hundred years, but errors in numerical methods can accumulate exponentially as time progresses, so that these numerical methods give virtually no information about the positions of the planets millions of year hence. The stability problem for the solar system was studied during the nineteenth century by many distinguished mathematicians, including Dirichlet, who claimed to have found a proof that the solar system is stable, but died before writing it down.

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King Oscar of Sweden offered a prize for a resolution of the stability problem, which was given to the French mathematician Henri Poincar´ e (1854-1912). Although Poincar´ e did not find a proof of the stability of the solar system, he did initiate many key ideas in the qualitative theory of differential equations, which has had an influence extending far beyond the solar system problem.

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