Reduced kinematic model of the human spine

According to the difficulty of marker based motion reconstruction of human torso movements a reduced kinematic formulation of the human spine is introduced, which describes the movement of the first thoracic vertebra in relation to the pelvis. The formulation is for implementation in a multibody model of the human body. As a result the formulation offers convenience in marker based motion capture and motion reconstruction via an optimization based kinematic approach and provides human-like motion characteristics of torso motion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relation between the mechanical properties of polymers and the kinetic flexibility of the macromolecules

Nuclear magnetic resonance combined with extension of the specimens in the NMR spectrometer on the reversible deformation range has been used to study the cooperative modes of motion of the macromolecules in polymers, specifically polycaprolactam and polyethylene terephthalate. For each of these polymers there are two temperatures near which the nature of the molecular motion changes sharply. At low temperatures there is a transition from independent hindered vibrations of the repeating units to correlated vibrations, as a result of which the chains in the amorphous regions acquire limited kinetic flexibility within the glassy state; at high temperatures there is a transition to segmental motion. The question of how changes in the modes of molecular motion are reflected in the macroscopic mechanical properties of polymers is examined.Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 24–29, January–February, 1971.     

Some Methods of Analysis of the Dynamic Systems with the Various Dissipation in Dynamics of a Rigid Body

In is well–known due to its complexity, the problem of the motion of a rigid body in an unbounded medium requires the introduction of certain simplifying restrictions. The main aim in this connection is to introduce hypotheses that would make it possible to study the motion of the rigid body separately from the motion of the medium in which the body is embedded. On the one hand, a similar approach was realized in the classical Kirchhoff problem on the motion of a body in an unbounded ideal incompressible fluid that undergoes an irrotational motion and is at rest at infinity. On the other hand, it is obvious that the above–mentioned Kirchhoff problem does not exhaust the possibilities of this kind of simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an Identity in Law for the Variance of the Brownian Bridge

An explanation of a duplication identity in law involving thevariances of the Brownian bridge and Brownian motion is given,with the help of an elementary transformation in time and spaceof a two-dimensional Brownian motion. 1991 Mathematics SubjectClassification 60J65.     

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.     

The rate of quasi sure convergence in the functional limit theorem for increments of a Brownian motion

We investigate the quasi sure convergence of the functional limit for increments of a Brownian motion. The rate of quasi sure convergence in the functional limit for increments of a d-dimensional Brownian motion is derived. The main tool in the proof is large deviation and small deviation for Brownian motion in terms of (r,p)-capacity.     

Lower Bounds for the Distribution of Suprema of Brownian Increments and Brownian Motion Normalized by the Corresponding Modulus Functions

The Lévy–Ciesielski construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process \((W_{t})_{t\in \left[ 0,T\right] }\) normalized by the global modulus function, for all positive \(\varepsilon \) and \(\delta \). Additionally, uniform results over \(\delta \) are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in establishing the results for the standard Brownian motion case.     

Coisotropic invariant tori of hamiltonian systems of the quasiclassical theory of motion of a conduction electron

We investigate quasiclassical equations of motion of a conduction electron in an electric and a magnetic field. It is shown that for homogeneous fields the motion can proceed along four-dimensional coisotropic invariant tori in six-dimensional phase space. By KAM-theory methods, the motion in weakly nonhomogeneous fields is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 346–351, March, 1990.     

Control of the motion of a helical body in a fluid using rotors

This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on SE(3) are obtained.     

Straight chains of interconnected mass points under the action of non-symmetric dry friction

The motion of a chain of interconnected equal material points placed on a rough straight line and connected by equal kinematical constraints is considered. It is supposed that the system experiences a small non-symmetric Coulomb dry frictional force acting from the line side upon each mass point and opposite to the direction of motion. The magnitude of this force depends on the direction of motion. To the equation of motion the procedure of averaging is applied. The expression for the stationary “on the average” velocity of the motion of the system as a single whole is found. The obtained theoretical results can be used for the design of worm-like vibration robots. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.     

Motion of a vortex filament in the half-space

A model equation for the motion of a vortex filament immersed in three-dimensional, incompressible and inviscid fluid is investigated as a preliminary attempt to model the motion of a tornado. We solve an initial–boundary value problem in the half-space, where we impose a boundary condition in which the vortex filament is allowed to move on the boundary.     

The dynamics of pyramidal bodies within the framework of the local interaction model

Two-dimensional inertial motion of pyramidal bodies in a medium is investigated, on the assumption that the force exerted by the medium on their surface is described by the local interaction model. Assuming unseparated flow around the bodies and small perturbations applied at the initial time to the parameters of rectilinear motion, an analytical solution is constructed of the problem of the two-dimensional motion of slender bodies with bases whose contour is a rhombus or a star consisting of four symmetrical cycles. It is shown that the solution provides the basis for a complete parameterc analysis of the dynamics of the body and for evaluating the forces and torques experienced by the body along its trajectory. A criterion for the stability of the body is found, using which, knowing the velocity, mass and position of the body's centre of gravity, one can determine the form of the perturbed motion of the pyramidal body. It is shown that the body shape is one of the most important factors affecting the stability of motion, and that, of all bodies with the same shape and position of the centre of mass, those with the least mass have the largest reserve of stability. The analytical results are confirmed by numerical solution of the Cauchy problem for the system of equations of motion obtained without the simplifying assumptions.     

Interpretation of the Evolution of the Homogeneous Markov System (or,equivalently, of the Embedded Markov Chain) as the Deformation of a Viscoelastic Medium. The 3-D Case

Every attainable structure of a continuous time homogeneous Markov chain (HMC) with n states, or of a closed Markov system with an embedded HMC with n states, or more generally of a Markov system driven by an HMC, is considered as a point-particle of ? ⁿ . Then, the motion of the attainable structure corresponds to the motion of the respective point-particle in ? ⁿ . Under the assumption that “the motion of every particle at every time point is due to the interaction with its surroundings”, ? ⁿ (and equivalently the set of the accosiated attainable structures of the homogeneous Markov system (HMS), or alternatively of the underlying embedded HMC) becomes a continuum. Thus, the evolution of the set of the attainable structures corresponds to the motion of the continuum. In this paper it is shown that the evolution of a three-dimensional HMS (n = 3) or simply of an HMC, can be interpreted through the evolution of a two-dimensional isotropic viscoelastic medium.     

The effect of the elasticity of an orbital tether system on the oscillations of a satellite

An orbital tether system, including a satellite (a rigid body), an elastic ponderable tether and a terminal load, is investigated. A mathematical model is obtained using Lagrange's equation of the second kind, which enables the plane translational motion of the centres of mass of the elements of the system and the rotational motion of the satellite and the tether to be investigated. It is shown that the equations of motion for the new independent variable, that is, the true anomaly angle, obtained on the assumption that the motion of the centre of mass of the system is independent of the relative motion of its elements, are an extension of the known mathematical models. The effect of the elasticity of the tether on the angular oscillations of the tether and the satellite is investigated. The model constructed can be used both to analyse of the deployment of a tether system as well as to investigate of the combined behaviour of a satellite and a tether about the natural centres of mass.     

Properties of the solution of the adjoint problem describing the motion of viscous fluids in a tube

For the Navier-Stokes equations, we study a solution invariant with respect to a oneparameter group and modeling a nonstationary motion of two viscous fluids in a cylindrical tube; the fluid layer near the tube wall can be viewed as a lubricant. The motion is due to a nonstationary pressure drop. We obtain a priori estimates for the velocities in the layers. We find a stationary state of the system and show that it is the limit state as t → ∞ provided that the pressure gradient in one of the fluids stabilizes with time. We solve the inverse problem of finding the pressure gradients and the velocity field from a known flow rate.     

Analysis of the steady motions of the tippe top

The motion of the tippe top on a horizontal plane is considered taking into account sliding friction within the Contensou model. The tippe top is modelled by two spherical segments rigidly joined by a rod directed along the common axis of symmetry of the segments. The dimensions of the spherical segments and the rod are chosen so that, as the axis of symmetry deviates from the upward vertical, the tippe top is supported on the plane at a point on one segment up to a certain critical value and at a point on the other segment at larger deviations (at points on both segments at the critical value). The motion of the tippe top is described by different equations in different regions of configuration space, and the motion is accompanied by impacts on the boundary of these regions. An effective potential of the system is constructed, and the type of its critical points is investigated. Poincaré–Chetayev bifurcation diagrams and generalized Smale diagrams are constructed for steady motions. Plots of the steady-state precessional motions have a discontinuity on the boundary between the regions indicated.     

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.

     

Existence of solutions for the equations

We introduce a concept of weak solution for a boundary value problem modelling the interactive motion of a coupled system consisting in a rigid body immersed in a viscous fluid. The fluid, and the solid are contained in a fixed open bounded set of R3. The motion of the fluid is governed by the incompresible Navier-Stokes equations and the standard conservation's laws of linear, and angular momentum rules the dynamics of the rigid body. The time variation of the fluid's domain (due to the motion of the rigid body) is not known apriori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary     

On the theory of the gyropendulum

Precession equations of motion of the gyropendulum relative to the accompanying Darboux trihedron /1/ and, also, precession equations of the gyropendulum motion relative to the geographic trihedron, considered in /2, 3/, are given a kinematic interpretation. Linear differential equations that define the gyropendulum behavior at finite deflection angles of the rotor axis from the vertical are established for arbitrary motions of its suspension point over the surface of the Earth. These equations have the form of kinematic equations of a solid body spherical motion in terms of Rodrigues-Hamilton parameters, and in the case of stationary base they are in agreement with equations established in /4/. The Liapunov stability ot the gyropendulum equations in both the finite Euler—Krylov angles and in the Rodrigues — Hamilton parameters is proved. Particular cases of integrability in quadratures of the gyropendulum precession equations at finite angles are indicated.