Abstract:
In this paper we shall define and study the angular momentum-energy space for the classical problem of plane-motions of a particle situated in a potential field of a central force. We shall present the angular momentum-energy space for some important cases.

Abstract:
In this paper we study a concept of mass-moment parameter which is the generalization of the mass and the moments of inertia of a continuous media. We shall present some interesting kinematical results in the hypothesis that a set of mass-moment parameters are conserved in a motion of a continuous media.

Abstract:
The paper present a model for the drag force between a resistive medium and a solid body using the hypothesis that the drag force is created by the adhesion of some particles of the resistive medium on the solid body's surface. The study focus on the mass evolution of the solid body.

Abstract:
We apply an algebraic method for studying the stability with respect to a set of conserved quantities for the problem of torque-free gyrostat. If the conditions of this algebraic method are not fulfilled then the Lyapunov stability cannot be decided using the specified set of conserved quantities.

Abstract:
We study the stability of the equilibrium points of a skew product system. We analyze the possibility to construct a Lyapunov function using a set of conserved quantities and solving an algebraic system. We apply the theoretical results to study the stability of an equilibrium state of a heavy gyrostat in the Zhukovski case.

Abstract:
We prove that the stability problem of a vertical uniform rotation of a heavy top is completely solved by using the linearization method and the conserved quantities of the differential system which describe the rotation of the heavy top.

Abstract:
We study the Lyapunov stability of a family of nongeneric equilibria with spin for underwater vehicles with noncoincident centers. The nongeneric equilibria belong to singular symplectic leaves that are not characterized as a preimage o a regular value of the Casimir functions. We find an invariant submanifold such that the nongeneric equilibria belong to a preimage of a regular value that involves sub-Casimir functions. We obtain results for nonlinear stability on this invariant submanifold.

Abstract:
On a Riemannian manifold $(M,g)$ we consider the $k+1$ functions $F_1,...,F_k,G$ and construct the vector fields that conserve $F_1,...,F_k$ and dissipate $G$ with a prescribed rate. We study the geometry of these vector fields and prove that they are of gradient type on regular leaves corresponding to $F_1,...,F_k$. By using these constructions we show that the cubic Morrison dissipation and the Landau-Lifschitz equation can be formulated in a unitary form.

Abstract:
In this paper we introduce the $\epsilon$ - revised system associated to a Hamilton - Poisson system. The $\epsilon$ - revised system of the rigid body with three linear controls is defined and some of its geometrical and dynamical properties are investigated.

Abstract:
We give a method which generates sufficient conditions for instability of equilibria for circulatory and gyroscopic conservative systems. The method is based on the Gramians of a set of vectors whose coordinates are powers of the roots of the characteristic polynomial for the studied systems. New instability results are obtained for general circulatory and gyroscopic conservative systems. We also apply this method for studying the instability of motion for a charged particle in a stationary electromagnetic field.