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On the multiplicity of brake orbits and homoclinics in Riemannian manifolds

Roberto Giambò, Fabio Giannoni, Paolo Piccione (2005)

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

Let M , g be a complete Riemannian manifold, Ω ⊂ M an open subset whose closure is diffeomorphic to an annulus. If ∂ ⁡ Ω is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in Ω ¯ = Ω ⁢ ⋃ ∂ ⁡ Ω starting orthogonally to one connected component of ∂ ⁡ Ω and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating...

On the notion of Jacobi fields in constrained calculus of variations

Enrico Massa, Enrico Pagani (2016)

Communications in Mathematics

In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of...

On the Singularities of the Newtonian two dimensional N-body Problem

Carlo Marchioro, Mario Pulvirenti (1983)

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

Si considera un sistema bidimensionale di N particelle interagenti tramite un potenziale di Newton o di Coulomb e si mostra che l’insieme delle condizioni iniziali che in un tempo finito possono condurre a delle singolarità possiede misura di Lebesgue nulla.

On the stability of Bravais lattices and their Cauchy–Born approximations

Thomas Hudson, Christoph Ortner (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze...

On the stability of Bravais lattices and their Cauchy–Born approximations*

Thomas Hudson, Christoph Ortner (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods,...

On the state observation and stability for uncertain nonlinear systems

Mohamed Ali Hammami (2000)

Kybernetika

In this paper, we treat the class of nonlinear uncertain dynamic systems that was considered in [3,2,1,4]. We consider continuous-time dynamical systems whose nominal part is linear and whose uncertain part is norm-bounded. We study the problems of state observation and obtaining stabilizing controller for uncertain nonlinear systems, where the uncertainties are characterized by known bounds.

On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics

Nalini Anantharaman (2004)

Journal of the European Mathematical Society

We study the zero-temperature limit for Gibbs measures associated to Frenkel–Kontorova models on ( ℝ d ) ℤ / ℤ d . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle–Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton–Jacobi equation. As the viscosity vanishes, the...

On weak solutions to the Lagrange-d'Alembert equation

Dmitry Treschev, Oleg Zubelevich (2013)

Applicationes Mathematicae

We consider nonholonomic systems with collisions and propose a concept of weak solutions to Lagrange-d'Alembert equations. Using this concept we describe the dynamics of collisions. Collisions of a rotating ball and a rough floor are considered.

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