Luigi Chierchia

Smooth prime integrals for quasi-integrable Hamiltonian systems

SummaryA Hamiltonian withN degrees of freedom, analytic perturbation of a canonically integrable strictly nonisochronous analytic Hamiltonian, is considered. We show the existence ofN functions on phase space and of classC∞ which are prime integrals for the perturbed motions on a suitable region whose Lebesgue measure tends to fill locally the phase space as the perturbation’s magnitude approaches zero. An application to the perturbations of isochronous nonresonant linear oscillators is given.RiassuntoSi considera un sistema hamiltoniano aN gradi di libertà, perturbazione analitica di un sistema analiticamente e canonicamente integrabile e strettamente non isocrono. Si mostra l’esistenza diN funzioni definite sullo spazio delle fasi e ivi di classeC∞ che sono integrali primi per il moto perturbato su opportune regioni la cui misura di Lebesgue tende a riempire localmente lo spazio delle fasi al tendere a zero della perturbazione. S’illustra un’applicazione alle perturbazioni di oscillatori isocroni non risonanti.

Construction of Analytic KAM Surfaces and Effective Stability Bounds

A class of analytic (possibly) time-dependent Hamiltonian systems withd degrees of freedom and the “corresponding” class of area-preserving, twist diffeomorphisms of the plane are considered. Implementing a recent scheme due to Moser, Salamon and Zehnder, we provide a method that allows us to construct “explicitly” KAM surfaces and, hence, to give lower bounds on their breakdown thresholds. We, then, apply this method to the HamiltonianH≡y2/2+ε(cosx+cos(x−t)) and to the map (y,x)→(y+ε sinx,x+y+ε sinx) obtaining, with the aid of computer-assisted estimations, explicit approximations (within an error of ∼10−5) of the golden-mean KAM surfaces for complex values of ε with |ε| less or equal than, respectively, 0.015 and 0.65. (The experimental numerical values at which such surfaces are expected to disappear are about, respectively, 0.027 and 0.97.) A possible connection between break-down thresholds and singularities in the complex ε-plane is pointed out.

A constructive theory of Lagrangian tori and computer-assisted applications

Perturbative techniques are among the most powerful tools in the theory of conservative dynamical systems. Besides giving finite time predictions (something well known to the astronomers of the eighteenth century), perturbation methods may be used to establish the existence of regular motions. H. Poincare used thoroughly such methods in his investigation in Celestial Mechanics [Po], obtaining, e.g., his celebrated results on periodic orbits for Hamiltonian systems. A more recent success of perturbation ideas is the so called “KAM (Kolmogorov [Ko]-Arnold [A1]-Moser [Mo1]) theory”, which ensures, under suitable smoothness assumptions, the survival under a small perturbation of “most” of the invariant maximal tori which foliate the phase-space of “integrable” conservative systems (see [B] for review and exhaustive references and [ChG] for recent developments).

Natural Boundaries for Area-Preserving Twist Maps

We consider KAM invariant curves for generalizations of the standard map of the form (x′, y′)=(x+y′, y+ɛf(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation θ↦θ+ω. In the complexɛ plane, natural boundaries of different shapes are found. In the complexθ plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 asɛ tends to the critical value.

Upper bounds on arnold diffusion times via Mather theory

We consider several Hamiltonian systems for which the existence of Arnolds mechanism for diffusion (whiskered tori, transition ladder, etc.) has been proven. By means of Mather theory we show that the diffusion time may be bounded by a power of the homoclinic splitting.

N-dimensional elliptic invariant tori for the planar (N + 1)-body problem !

For any

Rigorous estimates for a computer‐assisted KAM theory

N\geq 2

Compensations in small divisor problems

we prove the existence of quasi-periodic orbits lying on N-dimensional invariant elliptic tori for the planetary planar

HAMILTONIAN STABILITY OF SPIN-ORBIT RESONANCES IN CELESTIAL MECHANICS

(N+1)

Analytic Lagrangian tori for the planetary many-body problem

-body problem. For small planetary masses, such orbits are close to the limiting solutions given by the N planets revolving around the sun on planar circles. The eigenvalues of the linearized secular dynamics are also computed asymptotically. The proof is based on an appropriate averaging and KAM theory which overcomes the difficulties caused by the intrinsic degeneracies of the model. For concreteness, we focus on a caricature of the outer solar system.