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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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Convexity Methods in Hamiltonian Mechanics (Paperback, Softcover reprint of the original 1st ed. 1990)
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Convexity Methods in Hamiltonian Mechanics (Paperback, Softcover reprint of the original 1st ed. 1990)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 19
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In the case of completely integrable systems, periodic solutions
are found by inspection. For nonintegrable systems, such as the
three-body problem in celestial mechanics, they are found by
perturbation theory: there is a small parameter EURO in the
problem, the mass of the perturbing body for instance, and for EURO
= 0 the system becomes completely integrable. One then tries to
show that its periodic solutions will subsist for EURO -# 0 small
enough. Poincare also introduced global methods, relying on the
topological properties of the flow, and the fact that it preserves
the 2-form L~=l dPi 1\ dqi' The most celebrated result he obtained
in this direction is his last geometric theorem, which states that
an area-preserving map of the annulus which rotates the inner
circle and the outer circle in opposite directions must have two
fixed points. And now another ancient theme appear: the least
action principle. It states that the periodic solutions of a
Hamiltonian system are extremals of a suitable integral over closed
curves. In other words, the problem is variational. This fact was
known to Fermat, and Maupertuis put it in the Hamiltonian
formalism. In spite of its great aesthetic appeal, the least action
principle has had little impact in Hamiltonian mechanics. There is,
of course, one exception, Emmy Noether's theorem, which relates
integrals ofthe motion to symmetries of the equations. But until
recently, no periodic solution had ever been found by variational
methods.
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