# Sosnitskii S. P.

### On one atypical scheme of application of the second Lyapunov method

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1557-1563

The second Lyapunov method is applied to the analysis of stability of triangular libration points in a three-dimensional restricted circular three-body problem. It is shown that the triangular libration points are unstable.

### On the hill stability of motion in the three-body problem

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1434–1440

We consider the special case of the three-body problem where the mass of one of the bodies is considerably smaller than the masses of the other two bodies and investigate the relationship between the Lagrange stability of a pair of massive bodies and the Hill stability of the system of three bodies. We prove a theorem on the existence of Hill stable motions in the case considered. We draw an analogy with the restricted three-body problem. The theorem obtained allows one to conclude that there exist Hill stable motions for the elliptic restricted three-body problem.

### On the Lagrange Stability of Motion in the Three-Body Problem

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1137 – 1143

For the three-body problem, we study the relationship between the Hill stability of a fixed pair of mass points and the Lagrange stability of a system of three mass points. We prove the corresponding theorem establishing sufficient conditions for the Lagrange stability and consider a corollary of the theorem obtained concerning a restricted three-body problem. Relations that connect separately the squared mutual distances between mass points and the squared distances between mass points and the barycenter of the system are established. These relations can be applied to both unrestricted and restricted three-body problems.

### On the Stability of an Equilibrium State of Gyroscopic Coupled Systems

Ukr. Mat. Zh. - 2003. - 55, № 2. - pp. 255-263

We investigate the stability of an equilibrium state of gyroscopic coupled conservative systems in the case where the force function does not attain a local maximum in this state. We consider the situation where the gyroscopic coupling is weak with respect to a part of coordinates and strong with respect to the other part.

### On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1411–1416

For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system).

### On instability of the equilibrium state of nonholonomic systems

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 389–397

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.

### On instability of conservative systems with gyroscopic forces

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1422–1428

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.

### On the instability of lagrange solutions in the three-body problem

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1580-1585

We consider the relation between the Lyapunov instability of Lagrange equilateral triangle solutions and their orbital instability. We present a theorem on the orbital instability of Lagrange solutions. This theorem is extended to the planar*n*-body problem.

### On the gyroscopic stabilization of conservative systems

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1402-1408

We consider conservative systems with gyroscopic forces and prove theorems on stability and instability of equilibrium states for such systems. These theorems can be regarded as a generalization of the Kelvin theorem to nonlinear systems.

### Stability of equilibria of nonholonomic systems in a special case

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 440-447

### Stability of nonholonomic Chaplygin systems

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1100–1106

### Constructive instability of equilibrium of autonomous systems

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 95-101

### Action in hamilton's sense as an analogue of Lyapunov's function for natural systems

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 215–220

### Some cases of instability of equilibria of natural systems

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 124 – 127