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In the paper a necessary and sufficient condition for the stability of the equilibrium ([03]), barone-netto, a and cesar, m o study the stability of the solution (x, 'X PONTO', y, 'Y PONTO') = (0,0,0,0) of a mechanical system of two degrees of freedom 'X 2 PONTOS' = -xf (x) (x,y) 'PERTENCE' r x r 'Y 2 PONTOS' = -yg (x) f,g,'PERTENCE' 'C POT.2', f (0)>0, g (0)>0, by using a function l: [0, 'X BARRA IND.0] 'seta' 'r barra' ASSOCIATE TO THE SYSTEM, AND ALSO EXHIBIT A FIRST INTEGRAL TO THE SYSTEM, 'w barra': U 'contido' 'r pot.4' 'seta r' WHERE U IS A NEIGHBORHOOD OF THE ORIGIN, WHICH DECIDES THE STABILITY. THE AUTHORS CHARACTERIZE THE STABILITY THROUGH CONDITIONS ON L, AND ALSO SHOW THAT THE SOLUTION (X, 'x ponto', Y, 'y ponto')= (0,0,0,0) IS STABLE IF AND ONLY IF 'w barra' IS A LIAPUNOV FUNCTION FOR THE STABILITY. IN OUR WORK, WE GENERALIZE THIS RESULT FOR A MECHANICAL SYSTEM OF N+1 DEGREES OF FREEDOM OF THE TYPE 'x 2 pontos = -xf (x) (x,y) 'PERTENCE' r x 'R POT.N' 'Y 2 PONTOS = -G (X)Y G= ['g ind.Ij'] 'pertence' 'c pot.2', f (0)>0. As in [03], we characterize the stability in this case through of a explicit liapunov function