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Diffusion times and stability exponents for nearly integrable analytic systems

Pierre Lochak, Jean-Pierre Marco (2005)

Open Mathematics

For a positive integer n and R>0, we set B R n = x ∈ ℝ n | x ∞ < R . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian h r = 1 2 r 1 2 + . . . 1 2 r n - 1 2 + r n on 𝕋 n × B R n , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of 𝕋 n × B R n , and setting ε j : = h - H j C 0 ( V ) the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface,...

Dirac structures and dynamical r -matrices

Zhang-Ju Liu, Ping Xu (2001)

Annales de l’institut Fourier

The purpose of this paper is to establish a connection between various objects such as dynamical r -matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical r -matrices of simple Lie algebras 𝔤 , and prove that dynamical r -matrices are in one-one correspondence with certain Lagrangian subalgebras of 𝔤 ⊕ 𝔤 .

Discrete time markovian agents interacting through a potential

Amarjit Budhiraja, Pierre Del Moral, Sylvain Rubenthaler (2013)

ESAIM: Probability and Statistics

A discrete time stochastic model for a multiagent system given in terms of a large collection of interacting Markov chains is studied. The evolution of the interacting particles is described through a time inhomogeneous transition probability kernel that depends on the ‘gradient’ of the potential field. The particles, in turn, dynamically modify the potential field through their cumulative input. Interacting Markov processes of the above form have been suggested as models for active biological transport...

Distinguished Riemann-Hamilton geometry in the polymomentum electrodynamics

Alexandru Oană, Mircea Neagu (2012)

Communications in Mathematics

In this paper we develop the distinguished (d-) Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures and some geometrical Maxwell-like and Einstein-like equations) for the polymomentum Hamiltonian which governs the multi-time electrodynamics.

Divergence operators and odd Poisson brackets

Yvette Kosmann-Schwarzbach, Juan Monterde (2002)

Annales de l’institut Fourier

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, Δ , of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples...

Doubly asymptotic trajectories of Lagrangian systems and a problem by Kirchhoff

Maria Letizia Bertotti, Sergey V. Bolotin (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider Lagrangian systems with Lagrange functions which exhibit a quadratic time dependence. We prove the existence of infinitely many solutions tending, as t → ± ∞ , to an «equilibrium at infinity». This result is applied to the Kirchhoff problem of a heavy rigid body moving through a boundless incompressible ideal fluid, which is at rest at infinity and has zero vorticity.

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