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Luque, Alejandro
Publications (2 of 2) Show all publications
Haro, A. & Luque, A. (2019). A-posteriori KAM theory with optimal estimates for partially integrable systems. Journal of Differential Equations, 266(2-3), 1605-1674
Open this publication in new window or tab >>A-posteriori KAM theory with optimal estimates for partially integrable systems
2019 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 266, no 2-3, p. 1605-1674Article in journal (Refereed) Published
Abstract [en]

In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invariant tori for n-DOF Hamiltonian systems with additional n - d independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.

Place, publisher, year, edition, pages
National Category
Mathematical Analysis
urn:nbn:se:uu:diva-371106 (URN)10.1016/j.jde.2018.08.003 (DOI)000450909900025 ()
EU, European Research Council, 335079EU, Horizon 2020, MSCA 734557
Available from: 2018-12-19 Created: 2018-12-19 Last updated: 2018-12-19Bibliographically approved
Enciso, A., Luque, A. & Peralta-Salas, D. (2019). Stationary Phase Methods and the Splitting of Separatrices. Communications in Mathematical Physics, 368(3), 1297-1322
Open this publication in new window or tab >>Stationary Phase Methods and the Splitting of Separatrices
2019 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 368, no 3, p. 1297-1322Article in journal (Refereed) Published
Abstract [en]

Using stationary phase methods, we provide an explicit formula for the Melnikov function of the one and a half degrees of freedom system given by a Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably, the Melnikov function turns out to be computable using very little information on the separatrix and in the case of non-analytic systems. This is related to a priori stable systems coupled with low regularity perturbations. A natural physical application is to perturbations controlled by wave-type equations, so in particular we also illustrate this result with the motion of charged particles in a rapidly oscillating electromagnetic field. Quasi-periodic perturbations are discussed too.

Place, publisher, year, edition, pages
National Category
Other Physics Topics Mathematics
urn:nbn:se:uu:diva-385966 (URN)10.1007/s00220-019-03364-0 (DOI)000468174000009 ()
EU, European Research Council, 633152; 335079Knut and Alice Wallenberg Foundation, KAW 2015.0365
Available from: 2019-06-19 Created: 2019-06-19 Last updated: 2019-06-19Bibliographically approved

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