Lumped-Parameter Electromechanicsenergy in (3.1.22) and (3.1.23):i W(aw , z)a , X) (3.1.22)f" = w ) (3.1.23)axWe now differentiate (3.1.22) with respect to x and (3.1.23) with respect to L.Then, becauseaja. 8xMa2w, a2wathe reciprocity relation results:ai(, X) _ af(, x) (3.1.58)ax NThe process used in obtaining the reciprocity condition (3,1.58) shows thatthe condition is necessary for the system to be conservative. This samecondition can also be shown to be sufficient to ensure that the system isconservative. The proof requires a straightforward but involved integrationand is not carried out here primarily because it is a standard inclusion in somethermodynamics texts.*The reciprocity condition of (3.1.58) can be generalized to describe aconservative system with any number
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