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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Relative Equilibria of the Curved N-Body Problem (Hardcover, 2012 ed.)
Loot Price: R3,106
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Relative Equilibria of the Curved N-Body Problem (Hardcover, 2012 ed.)
Series: Atlantis Studies in Dynamical Systems, 1
Expected to ship within 12 - 17 working days
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The guiding light of this monograph is a question easy to
understand but difficult to answer: {What is the shape of the
universe? In other words, how do we measure the shortest distance
between two points of the physical space? Should we follow a
straight line, as on a flat table, fly along a circle, as between
Paris and New York, or take some other path, and if so, what would
that path look like? If you accept that the model proposed here,
which assumes a gravitational law extended to a universe of
constant curvature, is a good approximation of the physical reality
(and I will later outline a few arguments in this direction), then
we can answer the above question for distances comparable to those
of our solar system. More precisely, this monograph provides a
mathematical proof that, for distances of the order of 10 AU, space
is Euclidean. This result is, of course, not surprising for such
small cosmic scales. Physicists take the flatness of space for
granted in regions of that size. But it is good to finally have a
mathematical confirmation in this sense. Our main goals, however,
are mathematical. We will shed some light on the dynamics of N
point masses that move in spaces of non-zero constant curvature
according to an attraction law that naturally extends classical
Newtonian gravitation beyond the flat (Euclidean) space. This
extension is given by the cotangent potential, proposed by the
German mathematician Ernest Schering in 1870. He was the first to
obtain this analytic expression of a law suggested decades earlier
for a 2-body problem in hyperbolic space by Janos Bolyai and,
independently, by Nikolai Lobachevsky. As Newton's idea of
gravitation was to introduce a force inversely proportional to the
area of a sphere the same radius as the Euclidean distance between
the bodies, Bolyai and Lobachevsky thought of a similar definition
using the hyperbolic distance in hyperbolic space. The recent
generalization we gave to the cotangent potential to any number N
of bodies, led to the discovery of some interesting properties.
This new research reveals certain connections among at least five
branches of mathematics: classical dynamics, non-Euclidean
geometry, geometric topology, Lie groups, and the theory of
polytopes.
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