The Direct Method of Lyapunov

Abstract

Let us consider a problem of mechanics defined by the system of ordinary differential equations

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa % aaleaacaWGPbaabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaaqa % baGcdaqadaqaaiaadshacaGGSaGaamyCamaaBaaaleaacaaIXaaabe % aakiaacYcacaWGXbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6ca % caGGUaGaaiOlaiaacYcacaWGXbWaaSbaaSqaaiaad6gaaeqaaaGcca % GLOaGaayzkaaGaaiilaiaaykW7caWGPbGaeyypa0JaaGymaiaacYca % caaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGUbGaaiilaa % aa!53B8!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ {q_i} = {F_i}\left( {t,{q_1},{q_2},...,{q_n}} \right),\,i = 1,2,...,n, $$

(1.5.1)

and let us investigate the stability of a particular solution q ⁰ _i (t). For purposes of simplicity, we have omitted the influence of any parameters. The following variational equations are obtained by the well-known transformation (1.2) q _i = q ⁰ _i + ξ _i :

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS % baaSqaaiaadMgaaeqaaOGaeyypa0ZaaeWaaeaacqGHciITcaWGgbWa % aSbaaSqaaiaadMgaaeqaaOGaai4laiabgkGi2kaadghadaWgaaWcba % Gaam4AaaqabaaakiaawIcacaGLPaaadaqadaqaaiaadghadaqhaaWc % baGaamOAaaqaaiaaicdaaaGccaGGSaGaamiDaaGaayjkaiaawMcaai % abe67a4naaBaaaleaacaWGRbaabeaakiabgUcaRiabfA6agnaaBaaa % leaacaWGPbaabeaakmaabmaabaGaamiDaiaacYcacqaH+oaEdaWgaa % WcbaGaaGymaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiab % e67a4naaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaiaaykW7ca % WGPbGaaiilaiaadUgacqGH9aqpcaaIXaGaaiilaiaac6cacaGGUaGa % aiOlaiaacYcacaWGUbaaaa!65F0!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ {\xi _i} = \left( {\partial {F_i}/\partial {q_k}} \right)\left( {q_j^0,t} \right){\xi _k} + {\Phi _i}\left( {t,{\xi _1},...,{\xi _n}} \right)\,i,k = 1,...,n $$

(1.5.2)

.