The electrode potential-induced formation and dissipation dynamics of the hexagonal (''hex'') reconstruction on ordered Au(100) in perchloric and sulfuric acid electrolytes has been studied by means of in-situ scanning tunneling microscopy (STM). The real-space/real-time evolution of surface structures associated with the potential-dependent hex reversible (1 x 1) phase transition was examined on timescales down to about 1 s by acquiring STM images during appropriate potential sweeps and steps (dubbed here ''potentiodynamic STM''). Extensive hex domains can be formed by slow cooling following flame annealing and/or by holding the potential at values significantly below the potential of zero charge for the (1 x 1) surface. The sharp removal of the reconstruction seen voltammetrically, during positive-going potential sweeps, is accompanied by rapid (< 1 s) formation of arrays of ordered (1 x 1) clusters, created from the release of the about 24% additional gold atoms utilized in the (5 x 27) and related hex structures compared with the (1 x 1) substrate. These clusters are significantly, twofold, larger (approximately 4-6 nm) when formed in sulfuric acid electrolyte, due probably to an enhanced surface mobility in the presence of adsorbed sulfate. The reverse (1 x 1) --> hex transition at negative electrode charges is markedly slower. The hex domains appear initially as long thin (few atom-wide) strands, formed on (1 x 1) terraces by adatom diffusion primarily from cluster sites. This mechanism is augmented close to terrace edges by a ''wavefront-like'' motion of atomic rows. Further development of the hex domains occurs partly by aggregation of very thin hex strings, but primarily by a uniform broadening of thicker strands. The considerable prospects for utilizing potentiodynamic STM to explore local nanoscale processes associated with reconstruction and other potential-induced phase transitions are noted in the light of these findings.