Sensitivity Equations and Variational Equations

Abstract

We assume an undisturbed state q ⁰ _i , which satisfies the relationship

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaaca % WGXbWaa0baaSqaaiaadMgaaeaacaaIWaaaaOGaeyypa0JaamOramaa % BaaaleaacaWGPbaabeaakmaabmaabaGaamyCamaaDaaaleaacaWGRb % aabaGaaGimaaaakiaacYcacqaHXoqydaqhaaWcbaGaamOCaaqaaiaa % icdaaaGccaGGSaGaamiDaaGaayjkaiaawMcaauaabeqaceaaaeaaca % aMc8UaaGPaVlaaykW7caWGPbGaaiilaiaadUgacqGH9aqpcaaIXaGa % aiilaiaaikdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaad6gaae % aacaWGYbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGG % UaGaaiOlaiaacYcacaGGSaGaamyBaaaaaiaaw2haaaaa!5F29!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ \left. {q_i^0 = {F_i}\left( {q_k^0,\alpha _r^0,t} \right)\begin{array}{*{20}{c}} {\,\,\,i,k = 1,2,...,n} \\ {r = 1,2,...,,m} \end{array}} \right\} $$

(1.2.1)

in a general phase space R _n . The system of first-order differential equations q _i = F _i which correspond to a mechanical system and are basic to Eqs. (1.2.1), can, for example, be given by the set of canonical Hamilton differential equa­tions. The α ⁰ _r are parameters, and t is the time. Owing to perturbations, which as a special case should also consist of parameter changes, the perturbed state q ^S _i is obtained. If it is assumed that parameter changes cause α ⁰ _r to pass into α ^S _r = α ⁰ _r + β _r and that the structure of the differential Eqs.