arXiv subjects

Laurent Smoch

Publications and source records attributed to Laurent Smoch.

5 recordsLinked to original sources

arXiv2015

On the functional equation $\displaystyle α\bf{u}+\mathcal{C}\star(χ\bf{u})=\bf{f}$

We study in this paper the functional equation $\displaystyle α\mathbf{u}(t)+\mathcal{C}\star(χ\mathbf{u})(t)=\mathbf{f}(t)$ where $α\in\mathbb{C}^{d\times d}$, $\mathbf{u},\mathbf{f}:\mathbb{R}\rightarrow\mathbb{C}^d$, $\mathbf{u}$ being unknown. The term $\mathcal{C}\star(χ\mathbf{u})(t)$ denotes the discrete convolution of an almost zero matricial mapping $\mathcal{C}$ with discrete support together with the product of $\mathbf{u}$ and the characteristic function $χ$ of a fixed segment. This equation combines some aspects of recurrence equations and/or delayed functional equations, so that we may construct a matricial based framework to solve it. We investigate existence, unicity and determination of the solution to this equation. In order to do this, we use some new results about linear independency of monomial words in matrix algebras.

math.FA↗

arXiv2014

Discrete calculus of variation for homographic configurations in celestial mechanics

We provide in this paper the discrete equations of motion for the newtonian $n$-body problem deduced from the quantum calculus of variations (Q.C.V.) developed in \cite{Cre,CFT,RS1,RS2}. These equations are brought into the usual lagrangian and hamiltonian formulations of the dynamics and yield sampled functional equations involving generalized scale derivatives. We investigate especially homographic solutions to these equations that we obtain by solving algebraic systems of equations similar to the classical ones. When the potential forces are homogeneous, homographic solutions to the discrete and classical equations may be related through an explicit expansion factor that we provide. Consequently, perturbative equations both in lagrangian and hamiltonian formalisms are deduced.

math-ph↗

arXiv2011

Discrete calculus of variations for quadratic lagrangians

We develop in this paper a new framework for discrete calculus of variations when the actions have densities involving an arbitrary discretization operator. We deduce the discrete Euler-Lagrange equations for piecewise continuous critical points of sampled actions. Then we characterize the discretization operators such that, for all quadratic lagrangian, the discrete Euler-Lagrange equations converge to the classical ones.

math.OC↗

arXiv2011

Discrete Calculus of Variations for Quadratic Lagrangians. Convergence Issues

We study in this paper the continuous and discrete Euler-Lagrange equations arising from a quadratic lagrangian. Those equations may be thought as numerical schemes and may be solved through a matrix based framework. When the lagrangian is time-independent, we can solve both continuous and discrete Euler-Lagrange equations under convenient oscillatory and non-resonance properties. The convergence of the solutions is also investigated. In the simplest case of the harmonic oscillator, unconditional convergence does not hold, we give results and experiments in this direction.

math.OC↗

arXiv2011

Quadratic choreographies

This paper addresses the classical and discrete Euler-Lagrange equations for systems of $n$ particles interacting quadratically in $\mathbb{R}^d$. By highlighting the role played by the center of mass of the particles, we solve the previous systems via the classical quadratic eigenvalue problem (QEP) and its discrete transcendental generalization. The roots of classical and discrete QEP being given, we state some conditional convergence results. Next, we focus especially on periodic and choreographic solutions and we provide some numerical experiments which confirm the convergence.

math.OC↗