Stability of Motion
A. M. Liapunov
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In a celebrated Memoire dated 1892 A. M. Liapunov founded modern stability theory and provided a powerful technique to test for it. It is not generally known, however, that about the same time he wrote important complements. They have now been translated for the first time into a western language, and form the topic of the present volume. Academic Press deserves considerable credit for having undertaken this noteworthy scientific task.
We recall that a French translation of the Memoire appeared in 1907, in the Annales de Toulouse and was photoreproduced in 1949 by Princeton University Press as No. 17 of the Annals of Mathematics Studies. Thus the full contribution of Liapunov to the theory of stability is now available to the West.
This volume offers a welcome addition to the growing literature centering around Liapunov’s classical stability theory), and available to the non-Soviet world.
positive values o f t . Taking into account the existing relation between x,y , 5, and y , it is not difficult to deduce from here that the unperturbed motion is stable. 4. As we saw, when p is odd the unperturbed motion is unstable. We can show, however, that in this case it is somehow conditionally stable. Namely, we can show that it is stable with respect to the perturbations satisfying the condition by < 0. (6) To this end consider the equation y + Y(X, Y> = c, (7) obtained from (3) by
order. + + 11. Let us return now to our problem. Dealing with a case pertaining to the first category, we consider system (1) under our usual assumption that the function X vanishes for y = 0. Whatever the negative number a is, we can always reduce ourselves to the case a = - 1. In order to do this we have only to transform system (1) using the substitution = (.-u)-[1/2("-1)l = (-u)-w2("-1)l Yl. Suppose therefore, as in the previous paragraph, that a = - 1. In the cases in which we are
achieved by some transformation of system (1). 12 References its results concerning second-order systems. In reference  there is a communication that on December 4,1892, at the session of the Char’kov Mathematical Society, Liapunov reported a work with the same title as this manuscript and reference [ 11. There are in the manuscript so’me nonessential mistakes which are corrected in the text. The character of the corrections is mentioned in footnotes. Words, written shortly in the
I,+ l,q,s, Ip,q.s-l, I P + 2,q.s , Ip+2,q,s-19 IP.4,S (46) 3 Ip+4,q,s-l, Ip+l,q+n,s-l. (47) 164 Stability of Motion These equations can always be solved with respect to the quantities (46), since the determinant 1 k - b2 nb2 + 2(;- 1)n -(s k - 1) + 2 ( -~ l 1 ) ~-(s O- 1) I formed with the coefficients of these quantities, which can be brought into the form [k 2(s - 1)nI2 - (s - l)b2[k 2(s - l)n] n(s - 1)2b2,can not vanish when b2 < 4n, and k is different from zero.
S(0) = 0. (1 4 Different properties of the functions C(9) and S(9) are treated in detail in [ l ] and . It is shown, in particular, that C(9) and S(9) have the same period o > 0. The change of variables x = rC(9), * y = -rqS(9) In the original work Z , was used, which was a misprint. (1.3) Investigation of a Transcendental Case of the Theory of Stability of Motion 187 brings the system (0.1) to the form dr dt - = r q + ' R l ( r ,9) d9 dt - -- f dz dt - 1 - = AZ + rR2(r, z ,