scholarly journals Open problems, questions and challenges in finite- dimensional integrable systems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170430 ◽  
Author(s):  
Alexey Bolsinov ◽  
Vladimir S. Matveev ◽  
Eva Miranda ◽  
Serge Tabachnikov
Keyword(s):  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Theme Issue ◽  
Open Problems ◽  
Finite Dimensional

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference ‘Finite-dimensional Integrable Systems, FDIS 2017’ held at CRM, Barcelona in July 2017. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Symplectic invariants for parabolic orbits and cusp singularities of integrable systems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170424 ◽  
Author(s):  
Alexey Bolsinov ◽  
Lorenzo Guglielmi ◽  
Elena Kudryavtseva
Keyword(s):  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Normal Forms ◽  
Integrable Hamiltonian Systems ◽  
Theme Issue ◽  
New Techniques ◽  
Finite Dimensional ◽  
Parabolic Orbits ◽  
Symplectic Invariants ◽  
Cusp Singularities

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Asymptotic velocity for four celestial bodies

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170426
Author(s):  
Andreas Knauf
Keyword(s):  
Celestial Mechanics ◽  
Integrable Systems ◽  
Energy Surface ◽  
Initial Conditions ◽  
Theme Issue ◽  
Asymptotic Velocity ◽  
Pair Potentials ◽  
Finite Dimensional ◽  
Celestial Bodies ◽  
Almost All

Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here, we show for a class of pair potentials including the homogeneous ones of degree − α for α ∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Convexity of the moment map image for torus actions on b m -symplectic manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170420 ◽  
Author(s):  
Victor W. Guillemin ◽  
Eva Miranda ◽  
Jonathan Weitsman
Keyword(s):  
Symplectic Manifold ◽  
Integrable Systems ◽  
Symplectic Manifolds ◽  
Moment Map ◽  
Torus Action ◽  
Convexity Theorem ◽  
Theme Issue ◽  
Torus Actions ◽  
Finite Dimensional ◽  
The Moment

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a b m -symplectic manifold. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals A survey on polynomial in momenta integrals for billiard problems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170418 ◽  
Author(s):  
Misha Bialy ◽  
Andrey E. Mironov
Keyword(s):  
Constant Curvature ◽  
Integrable Systems ◽  
Theme Issue ◽  
Algebraic Version ◽  
Finite Dimensional ◽  
Short Survey ◽  
Magnetic Billiards ◽  
Algebraic Technique

In this paper, we give a short survey of recent results on the algebraic version of the Birkhoff conjecture for integrable billiards on surfaces of constant curvature. We also discuss integrable magnetic billiards. As a new application of the algebraic technique, we study the existence of polynomial integrals for the two-sided magnetic billiards introduced by Kozlov and Polikarpov. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals On the integrability of Birkhoff billiards

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170419 ◽  
Author(s):  
Vadim Kaloshin ◽  
Alfonso Sorrentino
Keyword(s):  
Integrable Systems ◽  
Theme Issue ◽  
Finite Dimensional

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Relations in the cohomology ring of the moduli space of flat SO (2 n  + 1)-connections on a Riemann surface

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170427
Author(s):  
Elisheva Adina Gamse ◽  
Jonathan Weitsman
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Integrable Systems ◽  
Marked Point ◽  
Cohomology Ring ◽  
Line Bundles ◽  
Chern Classes ◽  
Theme Issue ◽  
Gauge Transformations ◽  
Finite Dimensional

We consider the moduli space of flat SO (2 n  + 1)-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalizing a conjecture of Newstead. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals How low can vacuum energy go when your fields are finite-dimensional?

2019 ◽  
Vol 28 (14) ◽  
pp. 1944006
Author(s):  
ChunJun Cao ◽  
Aidan Chatwin-Davies ◽  
Ashmeet Singh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Degrees Of Freedom ◽  
Vacuum Energy ◽  
Quantum Field ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Klein Gordon ◽  
Dimensional Version

According to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this paper, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein–Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

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