arXiv:math/0611032v1 [math.SG] 2 Nov 2006 1 THE ε - REVISED SYSTEM OF THE RIGID BODY WITH THREE LINEAR CONTROLS Dan COM ˘ ANESCU, Mihai IVAN and Gheorghe IVAN Abstract. In this paper we introduce the ε - revised system associated to a Hamil- ton - Poisson system. The ε - revised system of the rigid body with three linear controls is defined and some of its geometrical and dynamical properties are investigated. 1 1 Introduction It is well known that many dynamical systems can be formulated using a Poisson structure (see for instance, R. Abraham and J. E. Marsden [1] and M. Puta [11]). The metriplectic systems was introduced by P. J. Morrison in the paper [8]. These systems combine both the conservative and dissipative systems. A metriplectic system is a differential system of the form ˙ x = P dH + gdC, where P is a Poisson tensor on a manifold M, g is a symmetric tensor of type (2, 0) on M, and H and C are two smooth functions on M with the additional requirements: (a) P dC = 0; (b) gdH =0 and (c) dC · gdC ≤ 0. The differential systems of the form ˙ x = P dH + gdC which satisfies only the conditions (a) and (b) are called almost metriplectic systems ( see Fish, [2]; Marsden, [7]; Ortega and Planas - Bielsa, [9] ). An interesting class of almost metriplectic systems are so-called the revised dynamical systems associated to Hamilton-Poisson systems (see Gh. Ivan and D. Opri¸ s, [5]). The control of the rotation rigid body is one of the problems with a large practical applicability. For this reason, in this paper we study the ε - revised dynamical system associated to the rigid body with three linear controls. 2 Almost metriplectic systems We start this section with the presentation of the concept of almost metriplectic manifold (see Ortega and Planas- Bielsa, [9]). Let M be a smooth manifold of dimension n and let C ∞ (M ) be the ring of smooth real-valued functions on M . A Leibniz manifold is a pair (M, [·, ·]), where [·, ·] is a Leibniz bracket on M , that is [·, ·]: C ∞ (M ) × C ∞ (M ) → C ∞ (M ) is a R - bilinear operation satisfying the following two conditions: (i) the left Leibniz rule: [f 1 · f 2 ,f 3 ]=[f 1 ,f 3 ] · f 2 + f 1 · [f 2 ,f 3 ] for all f 1 ,f 2 ,f 3 ∈ C ∞ (M ); 1 2000 Mathematical Subject Classification : 58F05. Key words and phrases: almost metriplectic system, ε- revised system, rigid body