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In this work, we derive an analytical procedure for obtaining amultidimensional washboard ratchet potential Uf for two-dimensional Josephsonjunctions array (TDJJA) with an applied magnetic field. The magnetic field isgiven in units of the quantum ux per plaquette or frustration of the form f = MN . The derivation is done under the assumption that the checkerboard patternground state or unit cell of a two-dimensional Josephson junctions array(TDJJA) is preserved under current bias. The RCSJ with a white noise termmodels the dynamics for each junction phase in the array. The multidimensionalpotential is the unique expression of the collective effects that emerge fromthe array in contrast to the single junction. First step in the procedure is towrite the equation for the phases for the unit cell by considering theconstraints imposed for the gauge invariant phases due to frustration. Secondlyand key idea of the procedure, is to perform a variable transformation from theoriginal systems of stochastic equations to a system of variables where thecondition for the equality of mixed second partial is forced via thePoincar\'e's theorem for differentials forms. This leads to a nonlinear matrixequation (equation (9) in the text), where the new coordinates variables xf areevaluated and where the potential exist. The transform matrix also permits thecorrect transformation of the original white noise terms of each junction tothe intensities in the xf variables. The commensurate symmetries of the groundstate pinned vortex lattice, leads to discrete symmetries to the part of thewashboard potential that does not contain a tilt due to the external biascurrent(equation(10) in the text). In this work we apply the procedure for theimportant cases f = 1 2 ; 1 3 .

Multidimensional Washboard Ratchet Potentials for Frustrated Two-Dimensional Josephson-Junctions Arrays on square lattices