  In this work, we derive an analytical procedure for obtaining a
multidimensional washboard ratchet potential Uf for two-dimensional Josephson
junctions array (TDJJA) with an applied magnetic field. The magnetic field is
given in units of the quantum ux per plaquette or frustration of the form f = M
N . The derivation is done under the assumption that the checkerboard pattern
ground state or unit cell of a two-dimensional Josephson junctions array
(TDJJA) is preserved under current bias. The RCSJ with a white noise term
models the dynamics for each junction phase in the array. The multidimensional
potential is the unique expression of the collective effects that emerge from
the array in contrast to the single junction. First step in the procedure is to
write the equation for the phases for the unit cell by considering the
constraints imposed for the gauge invariant phases due to frustration. Secondly
and key idea of the procedure, is to perform a variable transformation from the
original systems of stochastic equations to a system of variables where the
condition for the equality of mixed second partial is forced via the
Poincar\'e's theorem for differentials forms. This leads to a nonlinear matrix
equation (equation (9) in the text), where the new coordinates variables xf are
evaluated and where the potential exist. The transform matrix also permits the
correct transformation of the original white noise terms of each junction to
the intensities in the xf variables. The commensurate symmetries of the ground
state pinned vortex lattice, leads to discrete symmetries to the part of the
washboard potential that does not contain a tilt due to the external bias
current(equation(10) in the text). In this work we apply the procedure for the
important cases f = 1 2 ; 1 3 .
