Transformation functional integral identities

Finally, we note, that while in thermodynamics independent variables are changed via Legendre Transform, partition functions are changed via Laplace Transforms. In the thermodynamics limit, only the maximum term contributes to the integrals, and the two transformations become manifestly identical E Ethernio, V Vthermo and A Athermo- Thus, lu the thermodyuamic limit,... 

The Sqi and Sp do not represent virtual changes here as did the variations in the coordinates Sq in the variation of the action integral to describe a purely mathematical change in the system s trajectory between two times. Here they represent real infinitesimal changes in the coordinates and momenta. The generating function in this case will differ only by an infinitesimal amount from the one describing the identity transformation and hence can be written as... 

Economics in process control, 3, 10-11, 15, 26, 532-34 Environmental regulations, 3 Equal-percentage valve, 254, 255 Equations of state, 57 Equilibria, 56, 78 chemical, 56 phase, 56-57, 71, 75, 78 Error criteria (see Time integral criteria) Euler s identities, 131-32, 149 Experimental modeling, 45, 656 frequency response techniques, 668 process identification, 657-62 time constant determination, 228, 232 Exponential function, 130 approximations, 215-16 Laplace transform, 130 z-transform, 592... 

Theorem 5.4.3 (see [307]). The geodesic Sow in T M, where M is a two-dimensional Riemannian manifold, has an additional integral quadratic in momenta if and only if in any isothermic coordinates x, y the function X setting the metric satisSes equation (4), where R = R - 2iR2 is a holomorphic function of the variable z = X + ty, which is not identical zero and under transition to other isothermic coordinates is transformed in accordance with formula (7). 

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