Singularities in the dynamics of systems with non-ideal constraints

The fundamental problem of dynamics involving determining the generalised accelerations and reactions of constraints as a function of the applied forces is considered for mechanical systems with k ⩾ 1 non-ideal geometrical constraints. A relation is established between this problem and the analysis of the singularities of piecewise-smooth mappings of a space R^k into itself. For Coulomb-type friction, a criterion for there to be no paradoxes is obtained and it is shown that when k = 1 possible singularities are convolutions, while when k = 2 they are subdivided into a fold, a casp and a double fold. The well-known Painlevé-Klein example is considered in detail for cases of bilateral and unilateral contacts; a complete list of possible paradoxical situations is presented for the first time.     

Realizing constraints in the dynamics of systems with rolling

The possibilities of realizing constraints in the motion of systems containing kinematic pairs with small relative slips are investigated. It is shown that the limiting transition to infinite hardness of the contact forces (zero values of the slip velocities) can result in both classical non-holonomic systems and non-classical systems with primary Dirac constraints. The manifold defined by these non-classical constraints is not close to the manifold specified by the no-slip conditions in the general case. Situations in which particular constraints are realized are distinguished after examining the orders of magnitude of the terms on the right-hand and left-hand sides of the relations between the slip velocities and the generalized velocities.     

A cardinal number connected to the solvability of systems of difference equations in a given function class

Let ℝ^ℝ denote the set of real valued functions defined on the real line. A map D: ℝ^ℝ → ℝ^ℝ is said to be a difference operator if there are real numbers a _i, b _i (i = 1, …, n) such that (Dƒ)(x) = ∑ _i=1 ⁿ a _i ƒ(x + b _i) for every ƒ ∈ ℝ^ℝand x ∈ ℝ. By a system of difference equations we mean a set of equations S = {D _i ƒ = g _i: i ∈ I}, where I is an arbitrary set of indices, D _i is a difference operator and g _i is a given function for every i ∈ I, and ƒ is the unknown function. One can prove that a system S is solvable if and only if every finite subsystem of S is solvable. However, if we look for solutions belonging to a given class of functions then the analogous statement is no longer true. For example, there exists a system S such that every finite subsystem of S has a solution which is a trigonometric polynomial, but S has no such solution; moreover, S has no measurable solutions. This phenomenon motivates the following definition. Let be a class of functions. The solvability cardinal sc( ) of is the smallest cardinal number κ such that whenever S is a system of difference equations and each subsystem of S of cardinality less than κ has a solution in , then S itself has a solution in . In this paper we determine the solvability cardinals of most function classes that occur in analysis. As it turns out, the behaviour of sc( ) is rather erratic. For example, sc(polynomials) = 3 but sc(trigonometric polynomials) = ω ₁, sc({ƒ: ƒ is continuous}) = ω ₁ but sc({f : f is Darboux}) = (2 ^ω )⁺, and sc(ℝ^ℝ) = ω. We consistently determine the solvability cardinals of the classes of Borel, Lebesgue and Baire measurable functions, and give some partial answers for the Baire class 1 and Baire class α functions. Partially supported by Hungarian Scientific Foundation grants no. 49786,37758,F 43620 and 61600. Partially supported by Hungarian Scientific Foundation grant no. 49786.     

Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior

We consider the subcritical generalized Korteweg-de Vries equation

     

Variational equations with constraints

In this work we deal with a constrained variational equation associated with the usual weak formulation of an elliptic boundary value problem in the context of Banach spaces, which generalizes the classical results of existence and uniqueness. Furthermore, we give a precise estimation of the norm of the solution.     

Reducibility of nonlinear almost periodic systems of difference equations given on a torus

We establish sufficient conditions for a nonlinear system of difference equations x(t + 1) =x(t) + + P(x(t),t)+ to be reducible to the system y(t + 1) =y(t) + . Here, P(x, t) is a function 2-periodic in x_i(i = 1, ...,n) and almost periodic int with a frequency basis .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 404–412, April, 1994.This work was supported by Ukrainian State Committee on Science and Technology.     

The dynamics of systems with large gyroscopic forces and the realization of constraints

Lagrangian systems with a large multiplier N on the gyroscopic terms are considered. Simplified equations of motion of general form with holonomic constraints are obtained in the first approximation with respect to the small parameter ɛ = 1/N. The structure of the solutions of the precessional equations is examined.     

Reduction of systems of linear differential equations to jordan normal form

Summary Various methods are discussed of finding a non-singular matrix P such that PAP⁻¹=J, where J is theJordan normal form of A, with special reference to the problem of reducing the system of equations x=Ax to the form y=Jy, where y=Px. To Giovanni Sansone on his 70^th birth day.     

Reduction problems in multi-scale systems dynamics

The paper is devoted to the different aspects of mathematical modelling and qualitative analysis in dynamics of complex multi-scale systems that are generated by applied problems of engineering practice. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A general method for writing the dynamical equations of nonholonomic systems with ideal constraints

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.      

The dynamics of systems with rolling and gyroscopic systems with small generalized velocities and the realization of constraints

Approaches to the construction of mathematical models of systems with rolling and gyroscopic systems with dynamics characterized by the smallness of some of the generalized velocities are discussed. As a rule, a quasistatic approach is used in the modelling of such systems, within the limits of which the generalized accelerations corresponding to small generalized velocities are assumed to be equal to zero. Cases are indicated when the possibility, established by Kozlov, of obtaining the quasistatic equations of gyroscopic systems by the imposition of holonomic constraints is extended to systems with rolling. Additional conditions are formulated that enable one to estimate the error in the quasistatic equations of systems with rolling and gyroscopic systems. It is shown that they can be refined with respect to a small parameter, that is, the ratio of the characteristic values of the “small” and “finite” generalized velocities, using the Dirac formalism, based on an analysis of the constraints between the generalized coordinates and momenta of the system that arise on account of the degeneracy of its Lagrangian on changing to the quasistatic equations.     

Existence of a weak solution of Bogolyubov's hierarchical equations for infinite classical anharmonic systems with constraints

For systems of an infinite number of classical anharmonic oscillators with constraints one proves the existence of a weak solution of Bogolyubov's hierarchical equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 165–180, 1986.The author is grateful to O. A. Ladyzhenskaya for her interest in the paper.     

Reduction of the fundamental initial-boundary-value problems for Stokes equations to initial-boundary-value problems for parabolic systems of pseudodifferential equations

It is shown that an initial-boundary-value problem for Stokes' system, in which on the boundary one prescribes the vector field of velocities , or the stress field, or the normal component of the velocity and the tangential stresses, reduces to an initial-boundary-value problem for a system of the form , where the operator A contains a nonlocal term (the so-called singular Green operator). For the solutions of these problems, coercive estimates in the spaces W₂ ^{l, l/2} and also estimates of the norm of the resolving operator in W₂ ^r are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 37–48, 1987.     

Admissibility of simplified equations in the dynamics of gyroscopic systems

Based on the asymptotic approach of /1/, rigorous mathematical methods are used to single out some known simplified models from the theory of gyroscopic systems and to prove that they may legitimately be employed to solve problems in dynamics (including stability problems). The initial system is of the singularly perturbed type /2/. The use of methods from stability theory /3, 4/ yields conditions under which transition to a simplified (computational) model is permissible. Several papers have been devoted to the solution of such problems for singularly perturbed equations /5/ by methods of Lyapunov theory.