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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 € in the problem, the mass of the perturbing body for instance, and for € = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for € -# 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.

Inhaltsverzeichnis

Frontmatter

Chapter I. Linear Hamiltonian Systems

Abstract
Consider a system of m linear equations with continuous T -periodic coefficients:
$$ \dot x = M\left( t \right)x $$
(1)
where M (t) is a real m × m matrix, depending continuously on t ∈ ℝ such that:
$$ M\left( {t + T} \right) = M\left( t \right) $$
(2)
.
Ivar Ekeland

Chapter II. Convex Hamiltonian Systems

Abstract
We start from a duality pairing (X, X*, 〈·,·〉), that is, two real vector spaces X and X*, and a bilinear map (x,x*) → 〈x,x*〉 into ℝ which separates points:
$$ \begin{array}{*{20}c} {\forall x \in X,\exists x* \in X*:\,\left\langle {x,x*} \right\rangle \ne 0} \\ {\forall x* \in X*,\exists x \in X:\left\langle {x,x*} \right\rangle \ne 0} \\ \end{array} $$
.
Ivar Ekeland

Chapter III. Fixed-Period Problems: The Sublinear Case

Abstract
With this chapter, the preliminaries are over, and we begin the search for periodic solutions to Hamiltonian systems. All this will be done in the convex case; that is, we shall study the boundary-value problem
$$ \left\{ {\begin{array}{*{20}c} {\dot x = JH'\left( {t,x} \right)} \\ {x\left( 0 \right) = x\left( T \right)} \\ \end{array} } \right. $$
(1)
with H (t, ·) a convex function of x, going to +∞ when ∥ x ∥ → ∞.
Ivar Ekeland

Chapter IV. Fixed-Period Problems: The Superlinear Case

Abstract
We now consider the case when the Hamiltonian H is superquadratic, that is, behaves like ∥ xα for some α > 2. The dual action ψ can no longer be minimized (it is easy to check that inf ψ = −∞), so a subtler procedure is required. It will be described in the opening section.
Ivar Ekeland

Chapter V. Fixed-Energy Problems

Abstract
The fixed-energy problems are the most interesting (and the most difficult) in the theory, because of their geometric significance. Many are still unsolved, and we conclude this chapter by listing the most important ones.
Ivar Ekeland

Backmatter

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