Dragging along and invariant differentiation

by Dominic G. B. Edelen

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A physical system under coherent evolution will usually experience displacements of its constituent mass points; and consequently a description of the fields generated by the mass points must include the effects of these displacements. This Memorandum develops a succinct method whereby derivatives of general geometric fields can be evaluated at the deformed mass points in terms of quantities evaluated at the original undeformed mass points. A rather general invariant derivative is defined in the course of the analysis which has a number of interesting properties in its own right. With this derivative and the Lie diffference operator, relations between the dragged-along invariant derivative and the invariant derivative of the dragged-along field are derived. In all cases, the point transformations that give rise to the dragging along are considered as finite point transformations, and hence the use of the exponential operator is obviated. Under restriction to infinitesimal deformations, the results reduce to the commutation relations between invariant differentiation and Lie differentiation.

This report is part of the RAND Corporation Research memorandum series. The Research Memorandum was a product of the RAND Corporation from 1948 to 1973 that represented working papers meant to report current results of RAND research to appropriate audiences.

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