By C. Lanczos (auth.), R. C. Clark, G. H. Derrick (eds.)

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Sample text

91] with the boundary condition ... [92] The eigenvalue A. moves over to the boundary, which is in fact the source of the field inside the given domain. The orthogonality of the eigenfunctions ¢;(x) holds with respect to the boundary and the function on the boundary ¢(S) can be expanded into the orthonormal eigenfunctions ¢;(S): 24 C. LANCZOS 00

144] where e is a parameter that can be made as small as we wish. The same can be done equally with the other two coordinates and we may replace [ 144] by the more general ... [145] if we agree that we denote by xi the ensemble of the three Cartesian coordinates (xi> Yi• zi). Since later on we want to operate with the new coordinates xi and need not be afraid of misunderstandings if we omit the bar, we may express our transformation also as a replacement by saying that we replace xi by xi+eiX.

Let us assume that Lis a function of q 1 and til, but does not depend on t explicitly. Let us now transform the independent variable t to the new independent variable l = t+e-r This only means that we displace the zero-point of our clocks by a constant, which has no effect on the physical events, except that the q 1 become functions of l instead oft. But let us now again change -r to an added dynamical variable by making it a function of l. Now dt has to be replaced by dt = (l-ei)dl ... [152] ~ = (l+et) ~.