Some questions looking for answers in dynamical systems

Simó, Carles

doi:10.3934/dcds.2018267

Cited by 13 publications

(5 citation statements)

References 115 publications

(141 reference statements)

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“…As Cécile DeWitt-Morette said [11] "The role played by equilibrium points in the study of classical systems is analogous to the role played by classical paths in quantum systems." This is also in complete agreement with Carles Simó's idea on the fundamental role played by dynamical systems methods in many areas [32]. With this in mind it is very natural to consider the relevance of the variational equations along classical solutions for semiclassical expansions in quantum systems.…”

Section: Introductionsupporting

confidence: 83%

A Note on a Differential Galois Approach to Path Integrals

Morales-Ruiz

2019

Preprint

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We point out the relevance of the Differential Galois Theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself : if a finite dimensional complex analytical Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the Galois group of the variational equation around any integral curve must be abelian. A corollary of this result is that, for finite dimensional integrable Hamiltonian systems, the semiclassical approach is computable in closed form in the framework of the Differential Galois Theory. This explains in a very precise way the success of quantum semiclassical computations for integrable Hamiltonian systems.

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Section: Introductionsupporting

confidence: 83%

A Note on a Differential Galois Approach to Path Integrals

Morales-Ruiz

2019

Preprint

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“…The papers dealing with various special aspects of the KAM theory are exemplified by [8, 9, 25, 27-29, 33, 37, 40, 45, 47, 49]. Some open problems in the theory are listed and discussed in the works [23,30,44,50].…”

Section: Kronecker Torimentioning

confidence: 99%

Hamiltonian and reversible systems with smooth families of invariant tori

Sevryuk¹

2020

Preprint

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For various values of n, d, and the phase space dimension, we construct simple examples of Hamiltonian and reversible systems possessing smooth d-parameter families of invariant n-tori carrying conditionally periodic motions. In the Hamiltonian case, these tori can be isotropic, coisotropic, or atropic (neither isotropic nor coisotropic). The cases of non-compact and compact phase spaces are considered. In particular, for any N ≥ 3 and any vector ω ∈ R N , we present an example of an analytic Hamiltonian system with N degrees of freedom and with an isolated (and even unique) invariant N-torus carrying conditionally periodic motions with frequency vector ω (but this torus is atropic rather than Lagrangian and the symplectic form is not exact). Examples of isolated atropic invariant tori carrying conditionally periodic motions are given in the paper for the first time. The paper can also be used as an introduction to the problem of the isolatedness of invariant tori in Hamiltonian and reversible systems.

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“…that was first studied in [GJSS87] (see [Sim18] for later references). A phase portrait of this map, analogous to that shown for the standard map in Fig.…”

Section: Generalizations Of the Standard Mapmentioning

confidence: 99%

Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps

Sander,

Meiss

2020

Preprint

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Rotational invariant circles of area-preserving maps are an important and well-studied example of KAM tori. John Greene conjectured that the locally most robust rotational circles have rotation numbers that are noble, i.e., have continued fractions with a tail of ones, and that, of these circles, the most robust has golden mean rotation number. The accurate numerical confirmation of these conjectures relies on the map having a time reversal symmetry, and these methods cannot be applied to more general maps. In this paper, we develop a method based on a weighted Birkhoff average for identifying chaotic orbits, island chains, and rotational invariant circles that do not rely on these symmetries. We use Chirikov's standard map as our test case, and also demonstrate that our methods apply to three other, well-studied cases.

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Some questions looking for answers in dynamical systems

Cited by 13 publications

References 115 publications

A Note on a Differential Galois Approach to Path Integrals

A Note on a Differential Galois Approach to Path Integrals

Hamiltonian and reversible systems with smooth families of invariant tori

Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps

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