By P. Glansdorff
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Additional info for Thermodynamic Theory of Structure, Stability and Fluctuations
1851, On the oscillations of a suspension cable, Transactions ofthe Cambridge Philosophical Society, 9, 379-398. ), MacMillan and Co, London. S. , 1953, Modes of vibration of a suspended chain, Quarterly Journal ofMechanics and Applied Mathematics, 6, 273-285. C. , 1987, The role of nonlinear theories in transient dynamics analysis of flexible structures, Journal ofSound and Vibration, 119,487-508. Simo, lC. , 1991, A geometrically-exact rod model incorporating shear and torsion-warping deformation, International Journ al ofSolids and Structures, 27, 371-393.
1) Ricci symbol ei"I"k or ei .. N ) = when any two indices are equal, + I when (i" i2, ... , iN) are (I, 2, , N ) or an even permutation of (1, 2, -1 when(i" i2 , ... 105) , N) , ,iN ) are an odd permutation of (1, 2, .. , N) . 3. 110) ir js kt _ ri sj tk _I eijka a a - eijka a a - a mn 1erst' Proof : The asymmetry of Ricc i symbol gives For index (i , j , k ), eiik d/,a~a~ are asymmetric. n • Th us, IS (1 ,2,3) , 40 2 Tensor Analysis Similarly, the following formula can be proved. ir js eijka a a kt _ ri sj - eijka a a tk _I a mn 1erst ' - Therefore, this theorem is proved.
Consider another left decomposition as Y ·R=B=Y ·R, from which However, R ·R T =1 gives R- I =R T =iF _yT -(y-If =iF . , Y = yT and (y-I)T = y -I ). Taking inverse of tensor R-I gives Comparison of two expressions of R gives (y-I . Y- y . y -I) . R = O. Further, (V - I -Y- V · V-I) . R-RT = 0 => V-I . V = V . V-I. So V -V = V . V => V = V . From R = V-I -V . R , one obtains R = R . Therefore, the left decomposition is unique. Similarly, the uniqueness for the right decomposition can be proved .
Thermodynamic Theory of Structure, Stability and Fluctuations by P. Glansdorff