These invariants came from from the theory of bi-Hamiltnian systems on Lie algebras, so you may think of this subject as applications of integrable Hamiltonian systems in algebra.
(in cooperation with L.C. García-Naranjo, J. Montaldi)
We perform the reduction of the two-body problem in the two dimensional spaces of constant non-zero curvature and we use the reduced equations of motion to classify all relative equilibria (RE) of the problem and to study their stability. In the negative curvature case, the nonlinear stability of the stable RE is established by using the reduced Hamiltonian as a Lyapunov function. Surprisingly, in the positive curvature case this approach is not sufficient and the proof of nonlinear stability requires the calculation of Birkhoff normal forms and the application of stability methods coming from KAM theory. In both cases we summarize our results in the Energy-Momentum bifurcation diagram of the system.
We consider a vector field X meromorphic on a neighbourhood of a projective algebraic curve, closure of a particular integral curve Γ of X. The vector field X is integrable if there exist independent vector fields Y1, ..., Yl-1, X commuting pairwise with first integrals F1, ..., Fn-l. The Ayoul-Zung Theorem gives necessary conditions in terms of Galois groups for meromorphic integrability of X in a neighbourhood of Γ. Conversely, if these conditions are satisfied, we prove that if the first normal variational equation has a virtually diagonal monodromy group with non resonance and Diophantine properties, X is meromorphically integrable on a finite covering of a neighbourhood of Γ. We then prove the same, relaxing the non-resonance condition but adding an additional Galoisian condition, which in fine is implied by the non resonance hypothesis. Using the same strategy, we then prove a linearisability result near 0 for a time dependant vector field X with X(0)=0 for all t.
Toric actions and integrable systems have always been hand in hand in the symplectic realm. Liouville-Mineur-Arnold's theorem for integrable systems on a symplectic manifold guarantees that in a neighbourhood of their regular compact fibers a toric action exists. This toric action was already used by Duistermaat to extend local action-angle coordinates (Darboux-Caratheodory) to a neighbourhood of the fiber. Duistermaat's trick works pretty well for regular Poisson manifolds (Laurent-Miranda-Vanhaecke). In this talk I will find an extension of Duistermaat's trick to the singular setting (one of the action functions is no longer smooth). My excuse to work with singular symplectic manifolds (mainly bm-symplectic manifolds and folded-type symplectic manifolds) comes from several examples native to celestial mechanics where regularization transformations (Mc Gehee, Kustanheimo, etc.) yield such singularities and it is convenient to have the power of symplectic-type techniques close to the collision set/line at infinity.
In this 40-minute talk, I will give a general overview of this theory and I will present briefly a desingularization procedure for the singular symplectic structure which can be extended to desingularize also integrable systems and action-angle coordinates. This game can be played further to obtain KAM theorems in the singular setting.
Small perturbations imposed on an integrable nonlinear multifrequency oscillatory system cause a slow evolution. During this evolution the system may pass through resonant states. There are important phenomena related to such passages: capture into resonance and scattering on resonance. We will discuss the dynamics on long time intervals on which many passages through resonances occur.
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Optimal control problems provide a large class of examples for integrability properties. Indeed, thanks to the Pontryagin Maximum Principle, and given any (controlled) initial dynamics, one can see an optimal control problem as a Hamiltonian system - maybe, with singularities - on a cotangent bundle. We will study such systems, given by the controlled two body problem and the restricted three body problem, give some properties of their flows, and study the integrability of the minimum-time two-body problem via Moralès-Ramis techniques. This is a joint work with Jean-Baptiste Caillau and Thierry Combot.
In 1980, Michael Somos invented integer sequences that have later been popularized and generalized (among others) by David Gale. A certain mystery around this class of sequences is probably due to their relation with a wealth of different topics, such as: elliptic curves, continued fractions, and more recently with cluster algebras and integrable systems. It turns out that the properties of ``integrality'' and ``integrability'' are related not only phonetically!
I will also discuss extended Somos-4 and Somos-5 and more general Gale-Robinson sequences, and construct a great number of new integer sequences that also look quite mysterious; their integrability is yet to be understood...
Collisionless Boltzmann dynamics is a fruitful problem for stellar systems dynamics. The search of equilibria is usually based on two approaches: one relies on the existence of integrals of motion, the other tries to reproduce a given mass density profile.
In 1959, Michel Hénon focused on stellar orbital properties to introduce his fundamental isochrone potential. Such a potential is defined in spherically symmetric systems by the fact that, when it exists, the radial period of any test charge only depends on its energy and not on its angular momentum.
We will present a characterization and completion of the whole set of isochrone potentials. Transposing special relativity ideas to the isochrone context, we will discuss the nature of isochrony. This allows us to deeper understand general symmetries which appear in gravitation such as Kepler's Laws and Bertrand's Theorem.
Reference: A. Simon-Petit, J. Perez and G. Duval, Isochrony in radial 3D potentials : classification, interpretation and applications, submitted, 2017
Given any N-body configuration x0, and any normalized collision-free configuration a, for any positive energy h>0 we construct a hyperbolic motion γ, starting from x0 at time t=0 and asymptotic to a as t tends to infinity. The curve γ is constructed as a limit of free-time minimizers. It is a joint work with E. Maderna. The statement of the result is similar to a previous result for parabolic motions, but here the configuration a is not supposed to be central. The proof is a bit more involved.
We develop a general conceptual approach to the problem of existence of action-angle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions: anything which is preserved by the system is also preserved by the associated torus actions. This approach allows us to obtain, among other things: a) the shortest and most conceptual easy to understand proof of the classical Arnold--Liouville--Mineur theorem; b) basically all known results in the literature about the existence of action-angle variables in various contexts can be recovered in a unifying way, with simple proofs, using our approach; c) new results on action-angle variables in many different contexts, including systems on contact manifolds, systems on presymplectic and Dirac manifolds, action-angle variables near singularities, stochastic systems, and so on. Even when there are no natural action variables, our approach still leads to useful normal forms for dynamical systems, which are not necessarily integrable.
Reference: https://arxiv.org/abs/1706.08859