By A M Lyapunov
This e-book makes extra extensively available the textual content of Lyapunov's significant memoir of the final challenge of the steadiness of movement. Translated by way of A. T. Fuller (University of Cambridge), the paintings is now on hand for the 1st time within the English language, and marked the centenary of the Russian ebook within the overdue 1800s. together with a biography of Lyapunov and a entire bibliography of his paintings, this glorious quantity will end up to be of primary curiosity to all these thinking about the idea that of the steadiness of movement, limitations of balance, and with nonlinear dynamics.
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N). Thus, if we assume that system (23) is regular, which brings in the equality and if we note that because of Lemma V the sum L As cannot be greater than S, we must have Now, under the same assumption, we have S +S'=O. Hence, on referring to Lemma VII, we conclude that the characteristic number of the function e is equal to S ILPn dt - e fPkk dt + Ak. We therefore get (Lemma V) A. M. Lyapunov 566 and from this. by virtue of the above-mentioned equality. there results Ak + AI, ~ O. Now the sum Ak + AI, cannot be positive [Lemma V.
N) with constant coefficients. , n). ] Downloaded By: [University of Ottawa] At: 00:59 28 April 2010 (i On a general case of the differential equations of disturbed motion II. [A new type of series ordered according to the powers of the arbitrary constants] Let us now go back to equations (I). In only considering, as before, real values of t not less than a certain limit to, we shall suppose that all the coefficients p~ml' m' ..... ) are real, continuous and bounded functions of t. We shall further assume that we can find positive constants M and A such that the inequalities are satisfied for all the considered values of t.
Concerning systems of equations with this form, we can establish the following proposition. THEOREM. For the system of equations (23) to be regular, it is necessary and sufficient that the sum of the characteristic numbers of the functions fPS'f dt - and e fp,u dt e should be equal to zero for every value of s. First let us show that this condition is necessary. We have, for equations (23), the following system of n independent solutions: _ (I) fPI' dt x,-e _ , . , n), ;=2 (n) X,=X 2= ... =Xn _ I = 0 , Jp""dt x" =e .