Equations of motion | EPFL Graph Search

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics. Types There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dy

Stochastic partial differential equations driven by Lévy white noises

Thomas Marie Jean-Baptiste Humeau

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Lévy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Lévy measure have a positive absolute moment. In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Lévy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise. We then study the non-linear stochastic heat equation driven by a general type of Lévy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a càdlàg version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.

EPFL2017

Monte Carlo Modeling of Crystal Channeling at High Energies

Philippe Jean Schoofs

Charged particles entering a crystal close to some preferred direction can be trapped in the electromagnetic potential well existing between consecutive planes or strings of atoms. This channeling effect can be used to extract beam particles if the crystal is bent beforehand. Crystal channeling is becoming a reliable and efficient technique for collimating beams and removing halo particles. At the European Organization for Nuclear Research (CERN), the installation of silicon crystals in the Large Hadron Collider (LHC) is under scrutiny by the UA9 collaboration with the goal of investigating if they are a viable option for the collimation system upgrade. This thesis describes a new Monte Carlo model of planar channeling which has been developed from scratch in order to be implemented in the FLUKA code simulating particle transport and interactions. Crystal channels are described through the concept of continuous potential taking into account thermal motion of the lattice atoms and using Moliere screening function. The energy of the particle transverse motion determines whether or not it is trapped between the crystal planes while single Coulomb scattering on lattice atoms can lead to dechanneling. The volume capture and reflection applying to quasi-channeled particles are also modeled. Analogously to dechanneling, single scattering is used to determine the occurrence of volume capture. The parameters of the crystals, such as torsion or miscut, are described as well. For channeled particles, the suppression of electromagnetic and nuclear collisions is implemented. It is stronger for particles oscillating close to the center of the channel and is crucial for a correct evaluation of the rate of dechanneling without having recourse to the use of a macroscopic dechanneling length. The UA9-H8 experiment conducted at CERN aims at investigating new crystal physics as well as characterizing crystals that can be of interest in view of the implementation of crystal collimation at CERN, including in the LHC. This experiment uses silicon strip detectors situated on both sides of the crystal. Putting together upstream and downstream tracks in coincidence and matching at an identical fitted location on the crystal, it yields information about the deflections given to the beam population. Several runs from the UA9-H8 experiment are analyzed and compared to the model results. Channeling and dechanneling rates, as well as angular distributions at crystal exit are shown to be in a very encouraging agreement both for strip and quasi-mosaic crystals.

EPFL2014

Covariant formulation of relativistic mechanics

Miguel Alexandre Ribeiro Correia

Accretion disks surrounding compact objects, and other environmental factors, deviate satellites from geodetic motion. Unfortunately, setting up the equations of motion for such relativistic trajectories is not as simple as in Newtonian mechanics. The principle of general (or Lorentz) covariance and the mass-shell constraint make it difficult to parametrize physically adequate 4-forces. Here, we propose a solution to this old problem. We apply our framework to several conservative and dissipative forces. In particular, we propose covariant formulations for Hooke???s law and the constant force and compute the drag due to gravitational and hard-sphere collisions in dust, gas, and radiation media. We recover and covariantly extend known forces such as Epstein drag, Chandrasekhar???s dynamical friction, and Poynting-Robertson drag. Variable-mass effects are also considered, namely, Hoyle-Lyttleton accretion and the variable-mass rocket. We conclude with two applications: (1) The free-falling spring, where we find that Hooke???s law corrects the deviation equation by an effective anti???de Sitter tidal force and (2) black hole infall with drag. We numerically compute some trajectories on a Schwarzschild background supporting a dustlike accretion disk.