[en] The evolution of the eccentricity e and inclination i of the orbit of a planet of the terrestrial group as a result of collisions and encounters with bodies of the swarm during the process of accumulation is discussed in the paper. Encounters, being more numerous than collisions, had the main influence. Differential equations are derived describing the variation of the expected average values of e² and i² with the growth of the planet's mass. The rms eccentricity and inclination of the planet's orbit prove to be smaller than the corresponding quantities for bodies of the swarm by about √m-bar'/m times, where m-bar' is the average mass of bodies in the swarm and m is the planet's mass. The Fokker-Planck equation for the distribution density of the eccentricity and inclination of the orbit of a growing planet is solved. The problem of comparing the theoretical results with observations is discussed

[en] While planetary ring dynamical behavior is frequently characterized in terms of streamline deformations and motions, it is presently suggested that the standard formula yielding the streamlines is informed by geometric rather than orbital elements, upon taking planetary oblateness into account. The geometric elements are accordingly related to the orbital ones, in order to derive the differential equations governing the geometric elements' variations with time. 14 references

[en] Travelling fronts for the Fisher equation, the reaction-diffusion systems with diffusion induced destabilization and an introduction to the Conley index are briefly discussed. 47 refs

[en] Controllability with respect to certain variables. We will examine dynamic systems that can be described by ordinary differential equations. Where x element-of D improper-subset Rⁿ is the phase vector; u element-of U improper-subset R^m is the control vector. The latter is a bounded measureable function of time t, t element-of T = [0, ∞]. We assume that the regions D and U are convex and contain the coordinate origin. We also assume the function f to be a function of its arguments that is continuously differentiable a sufficient number of times. We divide the phase vector into two subvectors x^T = (x_α^T, x_β^T) (x_α element-of D_α improper-subset R^α, x_β element-of D improper-subset R^β) and we introduce the following definition for system (1.1)

No abstract available

[en] Numerical integration of Newton's equation in multiple dimensions plays an important role in many fields such as biochemistry and astrophysics. Currently, some of the most important practical questions in these areas cannot be addressed because the large dimensionality of the variable space and complexity of the required force evaluations precludes integration over sufficiently large time intervals. Improving the efficiency of algorithms for this purpose is therefore of great importance. Standard numerical integration schemes (e.g., leap-frog and Runge-Kutta) ignore the special structure of Newton's equation that, for conservative systems, constrains the force to be the gradient of a scalar potential. We propose a new class of open-quotes spatial interpolationclose quotes (SI) integrators that exploit this property by interpolating the force in space rather than (as with standard methods) in time. Since the force is usually a smoother function of space than of time, this can improve algorithmic efficiency and accuracy. In particular, an SI integrator solves the one- and two-dimensional harmonic oscillators exactly with one force evaluation per step. A simple type of time-reversible SI algorithm is described and tested. Significantly improved performance is achieved on one- and multi-dimensional benchmark problems. 19 refs., 4 figs., 1 tab

[en] A new mathematical model is developed for the plane-parallel motion of a system consisting of the frame of a vehicle and a steering system that controls the motion. The stability of straight-line motion is investigated. In most studies on the stability of the straight-line motion of a motor vehicle and a tractor-trailer unit it is assumed that in disturbed motion the driver keeps the steerable wheels in a neutral (unturned) position: the angle θ through which the wheels are turned about the axis of the steering knuckle pivot is zero. The validity of the assumption that θ = 0 can be argued in only one way, i.e., the torsional stiffness Κ of the steering system is fairly high: if Κ → ∞, then θ = 0. For finite values of Κ we have θ = 0. In that case the variable θ is determined by its differential equation, i.e., the number of degrees of freedom of the vehicle increases by one. The dynamic behavior of vehicles in this formulation is considered in this paper

[en] A new approach is described to the connection of wave amplitudes across the turning points and singular points of second-order, linear, analytic, ordinary differential equations which can describe the modulation of physical waves or oscillators. The general class of singular points thereby defined (refer to section 15) contains many irregular ones of greater complexity than have been accessible before; however, genuine coalescence of singular points is not considered here. The asymptotic connection formulae are shown to result directly from the branch structure of the singular point (refer to section 15), indeed, to a first approximation, they reflect merely the gross, local branch structure. The proof (refer to section 15) relates the local structure of the solutions at the singular point to the asymptotic wave structure by a limit process justified by bounds on the degree of irregularity of solution structure. 8 references

[en] In this paper we are interested in the solution of differential equations containing random forces. In order to do this we use a method of the functional derivative. We also describe the connection between the Fokker-Planck equation and the Langevin equation. In the last part of this paper we analyze various type of Langevin equation and a kinetic equation in order to obtain the well-known diffusion coefficient of charged particles in a stochastic magnetic field. (Author)

[en] Curve-fitting methods are usually used to obtain the exact solution to vibration problems in which allowance is made for dry (Coulomb) friction, but these methods permit determination of the laws of motion only in individual cases. The fact that the initial differential equations contain a piecewise-linear function characterizing dry friction makes it difficult to establish-and, thus to analyze-the general law governing vibratory motion for this case. As a result, dry friction is replaced by an equivalent viscous friction, and the corresponding areas of the hysteresis loops are equated. However, such a substitution cannot be justified in many cases, since dry and viscous friction differ in physical nature and differently affect the main characteristics of both free and forced vibrations. Moreover, the area of the hysteresis loop is proportional to the square of the amplitude in viscous friction but is proportional to the first power of the latter in dry friction. If the method of signum-function delay is used, then it becomes possible to determine the continuous laws of motion of such systems and establish the features of dry friction compared to viscous friction