Journal of Science and Arts Year 10, No. 2(13), pp. 223-230, 2010 THE STRUCTURAL INFLUENCE OF THE FORCES ON THE STABILITY OF DYNAMICAL SYSTEMS MIRCEA LUPU 1,2 1 Transilvania University of Brasov, Faculty of Mathematics and Informatics, 500091, Brasov, Romania 2 Academy of Romanian Scientists, 050094, Bucharest, Romania Abstract. In this paper consider the autonomous dynamical system linear or linearized with 2 degree of freedom. In the system of equation of 4th degree, appear the structure generalized forces: - the conservative forces, - the non-conservative forces, the dissipative forces, the gyroscopically forces. In the linear system, these forces from the different structural combinations can produce the stability or the instability of the null solution. In this way are known the theorems of Thomson - Tait - Cetaev (T-T-C) for the configurations . We will introduce the non conservative forces , studying the stability with the Routh - Hurwitz criterion or construct the Liapunov function, obtaining some theorems with practical applications. ) (q K , ) G ) ( q N ) (q D  ) (q G  ( , K D N Keywords: qualitative theory, stability, system structures, decomposition. 1. INTRODUCTION In this section study the structural influence of the terms blocks on the stability of the null solution for the bi-dimensional system or equations with fourth degree, which is a linear or linearized system in first approximation for the nonlinear system [2, 4].                    0 = 0 = 2 22 1 21 2 22 1 21 2 22 1 21 2 22 1 21 1 2 12 1 11 2 12 1 11 2 12 1 11 2 12 1 11 1 x n x n x g x g x c x c x k x k x x n x n x g x g x c x c x k x k x             (1) In this system the matrix blocks are representing respectively the conservative (elastic) forces, the resistance (amortization) forces, the gyroscoplically forces and the non conservative forces. The characteristic polynomial for the Routh - Hurwitz criterion will be [1, 7]: Nx x G x C Kx , , ,   0 = ) ( ) ( ) ( ) ( = ) ( 22 22 22 22 2 21 21 21 21 12 12 12 12 11 11 11 11 2 n k g c n k g c n k g c n k g c P                      (2) The system (1) can be bring to the canonic form and making abstraction by the constant negative factor the stated forces will be respectively side by the system : ) , ( 2 1 x x ) , ( 2 22 1 21 2 12 1 11 x k x k x k x k F   , ) , ( 2 22 1 21 2 12 1 11 x c x c x c x c C       , ) , ( 2 22 1 21 2 12 1 11 x g x g x g x g G       , ) , ( 2 22 1 21 2 12 1 11 x n x n x n x n N   . Regarding this system there are classical contributions of the Liapunov and the theorems of Thomson - Tait - Cetaev [4], Merkin [2] and Crandall [4]. Here we'll distinguish these results and we'll make other structural contribution by examples. In the matricial calculus are known the decomposition theorems of the squared matrix. Any squared matrix can be decomposing in a sum of one symmetric matrix and the other one asymmetric B A M  = . Where ) ( 2 1 = M M A   , ) ( 2 1 = M M B   . ISSN: 1844 – 9581 Mathematics Section