On instability of conservative systems with gyroscopic forces

Theorems on equilibrium instability of conservative systems with gyroscopic forces are proved. The theorems obtained are nonlinear analogs of the Kelvin theorem. The equilibrium instability of the Chaplygin nonholonomic systems is considered. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1422–1428, October, 1997.     

Asymptotic solutions of Lagrangian systems with gyroscopic forces

We consider Lagrangian systems in the presence of nondegenerate gyroscopic forces. The problem of stability of a degenerate equilibrium pointO and the existence of asymptotic solutions is studied. In particular we show that nondegenerate gyroscopic forces in general have, at least formally, a stabilizing effect whenO is a strict maximum point of the potential energy. It turns out that when we switch on arbitrary small nondegenerate gyroscopic forces, a bifurcation phenomenon arises: the instability properties ofO are transferred to a compact invariant set which collapses atO when the gyroscopic forces are switched off.Work supported by Russian Fund of Basic Research, the Italian Research Council (CNR) and the Italian Ministery of University (MURST)     

On instability of equilibrium figures of rotating viscous incompressible self-gravitating liquid not subjected to capillary forces

We prove that axially symmetric equilibrium figures of uniformly rotating viscous incompressible liquid are unstable, when the second variation of the energy functional can take negative values. We assume that there is no surface tension. __________ Translated from Problemy Matematicheskogo Analiza, No. 33, 2006, pp. 91–101.     

On the asymptotic equilibrium of a population system with migration

In the analysis of economic and social issues of a country (or any larger or smaller socio-economic unit) the demographic dynamics of the considered population often play a crucial role. Very current emergencies in this respect are e.g. ageing, longevity risk, state-run healthcare etc. Over the last decade migration between EU countries also became an important issue, and in recent years the uncontrolled migration from non-EU countries is also a major concern. Therefore, the better theoretical understanding of the evolutionary mechanism of age-classified populations interacting via migration, is a timely modelling-methodological task. This paper is a preliminary demographic methodological contribution to a further research in support of socio-economic modelling and decision making concerning migration issues.It is known that in the framework of the classical age-specific Leslie model, under simple demographic conditions, a closed population in the long term tends to an equilibrium age distribution. As the main theoretical result of the paper, a similar convergence is proved for a system of several populations with migration between them, and this long-term behaviour (convergence theorem) is extended to systems of sex-structured populations. Based on the latter model, medium term projections are also analysed concerning the effect of migration among countries on the development of the old-age dependency ratio (the proportion of pensioner age classes to active ones), which is an aggregate scalar indicator of ageing, a major concern in most industrialized countries. Illustrative simulation analysis is carried out with data from three European countries.     

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.     

On instability of the equilibrium state of nonholonomic systems

We establish a criterion of instability for the equilibrium state of nonholonomic systems, in which gyroscopic forces may dominate over potential forces. We show that, similarly to the case of holonomic systems, the evident domination of gyroscopic forces over potential ones is not sufficient to ensure the equilibrium stability of nonholonomic systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 389–397, March, 1999.     

Periodic solutions to second order nonautonomous differential systems with gyroscopic forces

Existence of periodic solutions to second order differential systems with gyroscopic forces is considered via variational methods, where a generalized Ahmad–Lazer–Paul type condition is used. We do not impose the condition that the gyroscopic forces are small.     

The inverse problem for Lagrangian systems with certain non-conservative forces

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.     

Optimal stabilization of an equilibrium position of a rigid body using rotors system with friction forces

The optimal stabilization of one class of equilibrium positions of a rigid body using rotors system with internal friction is studied. The performance index is an integral form of a quadratic function in the state and control variables. Both of the angular velocity and orientation variables are regulated. The optimal driving control moments which stabilize this position are obtained using the conditions of ensuring the optimal asymptotic stability of this position as non-linear functions of phase coordinates of the body. Some known results on the optimal control of equilibrium positions of a rigid body are improved and new results are obtained. Numerical simulation is presented.     

A proximal augmented Lagrangian method for equilibrium problems

Considering a recently proposed proximal point method for equilibrium problems, we construct an augmented Lagrangian method for solving the same problem in reflexive Banach spaces with cone constraints generating a strongly convergent sequence to a certain solution of the problem. This is an inexact hybrid method meaning that at a certain iterate, a solution of an unconstrained equilibrium problem is found, allowing a proper error bound, followed by a Bregman projection of the initial iterate onto the intersection of two appropriate halfspaces. Assuming a set of reasonable hypotheses, we provide a full convergence analysis.     

On small perturbations of a nonlinear periodic system with degenerate equilibrium

This paper considers the reducibility and existence of periodic solutions for a class of nonlinear periodic system with a degenerate equilibrium point under small perturbations. By introducing some parameter, we consider an equivalent periodic system. Then we prove that by an affine linear periodic transformation the parameterized periodic system is reducible to one with zero as an equilibrium. Topological degree theorem ensures that for some parameter the result can go back to the original system. Then, we obtain a small periodic solution.     

On the stability of the equilibrium of a linear mechanical system

Zusammenfassung Anschliessend an die Zieglerschen Arbeiten wird das allgemeine lineare dynamische System mit zwei Freiheitsgraden in bezug auf seine Stabilität untersucht. Es werden dabei nichtkonservative Feldkräfte, lineare Dämpfung und gyroskopische Kräfte in Betracht gezogen. die Anwendung des dynamischen Stabilitätskriteriums gibt einige Sätze, welche über die schon vonThomson undTait herrührenden Resultate hinausgehen. Es zeigt sich, dass in einem nichtkonservativen System ohne Dämpfung bei Anwesenheit gyroskopischer Kräfte das Gleichgewicht niemals stabil sein kann. Dieser Satz gilt für Systeme mit drei Freiheitsgraden nicht ohne Ausnahmen. Ein vonZiegler gefundenes Ergebnis, wonach eine Diskontinuität in der Stabilitätsbedingung auftreten kann, wird allgemein diskutiert.