First return time to the contact hyperplane for n-degree-of-freedom vibro-impact systems

International audienceThe paper deals with the dynamics of conservative $N$-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing orbits exhibit a contact occurrence with vanishing incoming velocity which generates mathematical difficulties. Such problems are commonly tackled through the definition of a Poincaré section and the attendant First Return Map. It is known that the First Return Time to the Poincaré section features a square-root singularity near grazing. In this work, a non-orthodox yet natural and intrinsic Poincaré section is chosen to revisit the square-root singularity. It is based on the unilateral condition but is not transverse to the grazing orbits. A detailed investigation of the proposed Poincaré section is provided. Higher-order singularities in the First Return Time are exhibited. Also, activation coefficients of the square-root singularity for the First Return Map are defined. For the linear and periodic grazing orbits from which bifurcate nonlinear modes, one of these coefficients is necessarily non-vanishing. The present work is a step towards the stability analysis of grazing orbits, which still stands as an open problem

A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole II: The Full System

We construct a scattering theory for the linearised Einstein equations on a Schwarzschild background in a double null gauge. We build on the results of Part I \cite{Mas20}, where we used the energy conservation enjoyed by the Regge--Wheeler equation associated with the stationarity of the Schwarzschild background to construct a scattering theory for the Teukolsky equations of spin $\pm2$. We now extend the scattering theory of Part I to the full system of linearised Einstein equations by treating it as a system of transport equations which is sourced by solutions to the Teukolsky equations, leading to Hilbert space-isomorphisms between spaces of finite energy initial data and corresponding spaces of scattering states under suitably chosen gauge conditions on initial and scattering data. As a corollary, we show that for a solution which is Bondi-normalised at both past and future null infinity, past and future linear memories are related by an antipodal map.Comment: 169 page

Transient fields of coherent synchrotron radiation in a rectangular pipe

We found an exact analytical solution of the wave equation for a transient electromagnetic field of synchrotron radiation in the frequency domain. The exact solution represents the field which consists of the coherent and incoherent components of synchrotron radiation and the space charge field of the particle beam moving in a bending magnet. The field in the time domain is gotten by numerically Fourier transforming the values of the field calculated using the exact solution. The beam has an arbitrary charge density and current density which satisfy the equation of continuity. The beam is moving in a perfectly conducting rectangular pipe which is uniformly curved in a semi-infinite bending magnet. The exact solution is not self-consistent, i.e., this is an exact expression of the field for a given beam current. We do not solve the equation of motion of the beam in the present paper. On the basis of the exact expression of the field found in the present study, we discuss the applicability and accuracy of the paraxial approximation which is sometimes used to calculate the field and spectrum of coherent or incoherent synchrotron radiation

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum