Quantum manifestations of Nekhoroshev stability

2016 ◽  
Vol 380 (39) ◽  
pp. 3167-3172 ◽  
Author(s):  
Daniele Fontanari ◽  
Francesco Fassò ◽  
Dmitrií A. Sadovskií
Keyword(s):  
Nekhoroshev Stability

scholarly journals Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems

1999 ◽  
Vol 172 ◽  
pp. 443-444
Author(s):  
Massimiliano Guzzo
Keyword(s):  
Degenerate System ◽  
Escape Time ◽  
Kam Theorem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Arbitrary Potential ◽  
Hamiltonian Perturbation Theory ◽  
Three Body ◽  
Nekhoroshev Stability ◽  
Degenerate Hamiltonian Systems

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

scholarly journals Nekhoroshev-Stability ofL4andL5in the Spatial Restricted Problem

1999 ◽  
Vol 172 ◽  
pp. 445-446 ◽  
Author(s):  
Giancarlo Benettin ◽  
Francesco Fassò ◽  
Massimiliano Guzzo
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Critical Mass ◽  
Kam Theory ◽  
Birkhoff Normal Form ◽  
Three Body Problem ◽  
Arnold Diffusion ◽  
Convexity Assumption ◽  
Three Body ◽  
Nekhoroshev Stability

The Lagrangian equilateral pointsL4andL5of the restricted circular three-body problem are elliptic for all values of the reduced massμbelow Routh’s critical massμR≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekhoroshev-stability’: denoting byda convenient distance from the equilibrium point, one asks whetherfor any small єe > 0, with positiveaandb. Until recently this problem, as more generally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus onμ(see e.g .Giorgilli, 1989). Our aim was instead considering all values ofμup toμR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian exists and satisfies a ‘quasi-convexity’ assumption (Fassòet al, 1998; Guzzoet al, 1998; Niedermann, 1998).

Nekhoroshev-Stability of Elliptic Equilibria of Hamiltonian Systems

1998 ◽  
Vol 197 (2) ◽  
pp. 347-360 ◽  
Author(s):  
Francesco Fassò ◽  
Massimiliano Guzzo ◽  
Giancarlo Benettin
Keyword(s):  
Hamiltonian Systems ◽  
Nekhoroshev Stability

scholarly journals Nekhoroshev Stability for the Dirichlet Toda Lattice

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 506
Author(s):  
Andreas Henrici
Keyword(s):  
Toda Lattice ◽  
Dirichlet Boundary Conditions ◽  
Periodic Lattice ◽  
Fixed Point Set ◽  
The Family ◽  
Theory Result ◽  
Point Set ◽  
Probabilistic Character ◽  
Nekhoroshev Stability ◽  
Stability Results

In this work, we prove a Nekhoroshev-type stability theorem for the Toda lattice with Dirichlet boundary conditions, i.e., with fixed ends. The Toda lattice is a member of the family of Fermi-Pasta-Ulam (FPU) chains, and in view of the unexpected recurrence phenomena numerically observed in these chains, it has been a long-standing research aim to apply the theory of perturbed integrable systems to these chains, in particular to the Toda lattice which has been shown to be a completely integrable system. The Dirichlet Toda lattice can be treated mathematically by using symmetries of the periodic Toda lattice. Precisely, by treating the phase space of the former system as an invariant subset of the latter one, namely as the fixed point set of an important symmetry of the periodic lattice, the results already obtained for the periodic lattice can be used to obtain analogous results for the Dirichlet lattice. In this way, we transfer our stability results for the periodic lattice to the Dirichlet lattice. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of related theorems, and the lattice with fixed ends is more important for applications than the periodic one.

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