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2013
Impact Factor

# Luigi Chierchia

Largo San Leonardo Murialdo 1, I - 00146 Roma, Italy
Dip di Matematica e Fisica, Università Roma Tre

## Publications:

 Chierchia L., Koudjinan C. E. V. I. Arnold’s ''Global'' KAM Theorem and Geometric Measure Estimates 2021, vol. 26, no. 1, pp.  61-88 Abstract This paper continues the discussion started in [10] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit ''global'' Arnold's KAM theorem, which yields, in particular, the Whitney conjugacy of a non-degenerate, real-analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov set are provided in case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the $d$-torus and (B) a domain with $C^2$ boundary times the $d$-torus. All constants are explicitly given. Keywords: nearly-integrable Hamiltonian systems, perturbation theory, KAM theory, Arnold’s scheme, Kolmogorov set, primary invariant tori, Lagrangian tori, measure estimates, small divisors, integrability on nowhere dense sets, Diophantine frequencies Citation: Chierchia L., Koudjinan C. E.,  V. I. Arnold’s ''Global'' KAM Theorem and Geometric Measure Estimates, Regular and Chaotic Dynamics, 2021, vol. 26, no. 1, pp. 61-88 DOI:10.1134/S1560354721010044
 Chierchia L., Koudjinan C. V. I. Arnold's “Pointwise” KAM Theorem 2019, vol. 24, no. 6, pp.  583-606 Abstract We review V.I. Arnold's 1963 celebrated paper [1] Proof of A.N. Kolmogorov's Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold's scheme, one can get ''sharp'' asymptotic quantitative conditions (as $\varepsilon \to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed. Keywords: Nearly-integrable Hamiltonian systems, KAM theory, Arnold's Theorem, small divisors, perturbation theory, symplectic transformations Citation: Chierchia L., Koudjinan C.,  V. I. Arnold's “Pointwise” KAM Theorem, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 583-606 DOI:10.1134/S1560354719060017
 Chierchia L. Meeting Jürgen Moser 2009, vol. 14, no. 1, pp.  5-6 Abstract Citation: Chierchia L.,  Meeting Jürgen Moser, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 5-6 DOI:10.1134/S156035470901002X
 Chierchia L. Kolmogorov’s 1954 Paper on Nearly-Integrable Hamiltonian Systems 2008, vol. 13, no. 2, pp.  130-139 Abstract Following closely Kolmogorov’s original paper [1], we give a complete proof of his celebrated Theorem on perturbations of integrable Hamiltonian systems by including few "straightforward" estimates. Keywords: Kolmogorov’s theorem, KAM theory, small divisors, Hamiltonian systems, perturbation theory, symplectic transformations, nearly-integrable systems Citation: Chierchia L.,  Kolmogorov’s 1954 Paper on Nearly-Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 130-139 DOI:10.1134/S1560354708020056
 Celletti A., Chierchia L. Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics 1998, vol. 3, no. 3, pp.  107-121 Abstract Birkhoff periodic orbits associated to spin-orbit resonances in Celestial Mechanics and in particular to the Moon–Earth and Mercury–Sun systems are considered. A general method (based on a quantitative version of the Implicit Function Theorem) for the construction of such orbits with particular attention to "effective estimates" on the size of the perturbative parameters is presented and tested on the above mentioned systems. Lyapunov stability of the periodic orbits (for small values of the perturbative parameters) is proved by constructing KAM librational invariant surfaces trapping the periodic orbits. Citation: Celletti A., Chierchia L.,  Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 107-121 DOI:10.1070/RD1998v003n03ABEH000084