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I show how there is an ambiguity in how one treats auxiliary variables ingauge theories including general relativity cast as 3 + 1 geometrodynamics.Auxiliary variables may be treated pre-variationally as multiplier coordinatesor as the velocities corresponding to cyclic coordinates. The latter treatmentworks through the physical meaninglessness of auxiliary variables' valuesapplying also to the end points (or end spatial hypersurfaces) of thevariation, so that these are free rather than fixed. [This is also known asvariation with natural boundary conditions.] Further principles of dynamicsworkings such as Routhian reduction and the Dirac procedure are shown to haveparallel counterparts for this new formalism. One advantage of the new schemeis that the corresponding actions are more manifestly relational. While theelectric potential is usually regarded as a multiplier coordinate and Arnowitt,Deser and Misner have regarded the lapse and shift likewise, this paper'sscheme considers new {\it flux}, {\it instant} and {\it grid} variables whosecorresponding velocities are, respectively, the abovementioned previously usedvariables. This paper's way of thinking about gauge theory furthermore admitsinteresting generalizations, which shall be provided in a second paper.

New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split