On the fall of a heavy rigid body in an ideal fluid

We consider a problem about the motion of a heavy rigid body in an unbounded volume of an ideal irrotational incompressible fluid. This problem generalizes a classical Kirchhoff problem describing the inertial motion of a rigid body in a fluid. We study different special statements of the problem: the plane motion and the motion of an axially symmetric body. In the general case of motion of a rigid body, we study the stability of partial solutions and point out limiting behaviors of the motion when the time increases infinitely. Using numerical computations on the plane of initial conditions, we construct domains corresponding to different types of the asymptotic behavior. We establish the fractal nature of the boundary separating these domains.     

Peng’s maximum principle for a stochastic control problem driven by a fractional and a standard Brownian motion

We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2. We apply an anticipative Girsanov transformation to transform the system into another one, driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion. We derive a maximum principle and the associated stochastic variational inequality, which both are generalizations of the classical case.     

Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.     

The optimal quasi-stationary motion of a vibration-driven robot in a viscous medium

We consider a rectilinear quasi-stationary motion of a two-mass system in a viscous medium. The motion of the system as a whole occurs due to periodic movements of the internal mass relatively to the shell. The problem is to describe the law of motion of the internal mass that provides the minimum energy consumption with a specified average velocity of the shell. We propose an algorithm for solving the problem with any law of the resistance of the medium. We obtain the energy-optimal law of motion of a spherical shell in a viscous liquid.     

Shell dynamics in the presence of a central body

We construct an idealized spherically symmetric relativistic model of an exploding object in the framework of the theory of surface layers in general relativity and match a Vaidya solution for a radially radiating star to another Vaidya solution through a thin spherical shell. We reduce the equations of motion and the radiation density of the Vaidya solution given by the matching conditions to a first-order system and analyze the general characteristics of the motion. We use a post-Newtonian approximation to find the equation of motion of a spherically symmetric radiating shell moving in a central gravitational potential.     

Nekhoroshev type stability results for Hamiltonian systems with an additional transversal component

We prove exponential stability theorems of Nekhoroshev type for motion in the neighbourhood of an elliptic fixed point in Hamiltonian systems having an additional transverse component of arbitrary dimension, under certain conditions on this transverse component. We consider both the cases (i) of a strongly constrained motion, and (ii) of a weak perturbation.     

Process of establishing a plane-wave system on ice cover over a dipole moving uniformly in an ideal fluid column

We consider a planar evolution problem for perturbations of the ice cover by a dipole starting its uniform rectilinear horizontal motion in a column of an initially stationary fluid. Using asymptotic Fourier analysis, we show that at supercritical velocities, waves of two types form on the water–ice interface. We describe the process of establishing these waves during the dipole motion. We assume that the fluid is ideal and incompressible and its motion is potential. The ice cover is modeled by the Kirchhoff–Love plate.     

On the solution of the equations of perturbed motion of an object-parachute system

We propose a method of finding the solutions of the equations of perturbed motion of an object-parachute system in analytic form. We perform an analysis of the roots of the characteristic equation of the linearized system. On the basis of the analysis we determine all the solutions of the equations of perturbed motion of the object-parachute system. We exhibit conditions under which the unperturbed motion (descent with slipping) is asymptotically stable.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 41–46.     

Small vibrations of an elastic conductor in a magnetic field

In this paper we study the motion of an elastic conducting wire in a magnetic field. The motion of the conductor induces a current in the wire (Faraday's law) which, in turn produces a force on the wire. We consider the linear equation obtained by linearizing the resulting equations of motion about an equilibrium solution. This is a hyperbolic partial differential equation with a non-local term. We prove existence and uniqueness of a weak solution of an initial–boundary value problem for this equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A note on stability of spacecrafts and underwater vehicles

A Hamilton‐Poisson system is an approach for the motion of a spacecraft around an asteroid or for the motion of an underwater vehicle. We construct a coordinate chart on the symplectic leaf which contains a specific generic equilibrium point and we establish stability conditions for this equilibrium point.     

On a Cameron–Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

We present a Cameron–Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion.     

Operator methods in the problem of perturbed motion of a rotating body partially filled with liquid

We apply operator methods to the investigation of an initial boundary-value problem which describes the perturbed motion of a body with cavity partially filled with an ideal liquid relative to the uniform rotation of this system about a fixed axis. We prove the existence and uniqueness of generalized solutions with finite energy and establish a sufficient condition for the stability of motion and some properties of the spectrum of the problem under consideration.     

Brownian motion on the continuum tree

Summary We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process.     

Ballistic motion of a tracer particle coupled to a Bose gas

We study the motion of a heavy tracer particle weakly coupled to a dense interacting Bose gas exhibiting Bose–Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations. We derive the effective dynamics of the tracer particle, which is described by a non-linear integro-differential equation with memory, and prove that if the initial speed of the tracer particle is below the speed of sound in the Bose gas the motion of the particle approaches an inertial motion at constant velocity at large times.     

The motion of a solid with large deformations

We study the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, plates… Therefore it is wise to choose a third-gradient theory for the body. Stretch matrix of the polar decomposition has to be symmetric. This is an internal constraint which introduces a reaction stress in the Piola–Kirchhoff–Boussinesq stress. We prove that there exists a motion that satisfies the complete equations of Mechanics in a convenient variational framework. This motion is local-in-time because it may be interrupted by crushing, resulting in a discontinuity of velocity with respect to time, i.e., an internal collision.     

Propagation of disturbances in a flexible extensible filament under longitudinal accelerated motions in a fluid

We seek a solution of the linearized equation of motion of a flexible extensible filament in a fluid in the form of an expansion in eigenfunctions of a boundary-value problem. For a uniformly accelerated motion and for motion accelerated according to a hyperbolic tangent law we find the exact solutions. For other forms of accelerated motion we propose a numerical solution of the initial inhomogeneous problem. We carry out an analysis of the solutions obtained. It is found that the first peak of the tension depends only weakly on the resistance of the fluid, but strongly on the acceleration parameters. The natural vibrations damp out more rapidly both as the resistance increases and as the acceleration increases. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 104–110.     

Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: II

We study the rectilinear motion of a mass point with impacts against a stopper. The motion between the impacts is described by a second-order ordinary differential equation with a parameter. The impact recovery coefficient also depends on the parameter of the vibroimpact system. We describe a bifurcation that leads to the generation of a Smale horseshoe in a parametric neighborhood of the chatter phenomenon. We prove the existence of an invariant set described by symbolic dynamics.     

Corrigendum to “Third-order partial differential equations arising in the impulsive motion of a flat plate” [Commun Nonlinear Sci Numer Simulat 2009;14:2629–36]

We correct an oversight in our recently published paper Van Gorder and Vajravelu [Third-order partial differential equations arising in the impulsive motion of a flat plate. Commun Nonlinear Sci Numer Simulat 2009;14:2629–36], regarding the correct form of the boundary condition in Stokes’ first problem for the impulsive motion of an infinite flat plate. We present the corrected solution to the flow problem.     

Chaos in vibroimpact systems with one degree of freedom in a neighborhood of chatter generation: I

We study the rectilinear motion of a mass point with impacts against a delimiter. The motion between the impacts is described by a second-order ordinary differential equation with a parameter. The impact recovery coefficient also depends on the parameter of the vibroimpact system. We describe a bifurcation that leads to the generation of a Smale horseshoe in a parametric neighborhood of the chatter phenomenon. We prove the existence of an invariant set described by symbolic dynamics.