The gauge problem of monopole dynamics is studied in SU(2) lattice gauge theory. We study first the Abelian and monopole contributions to the static potential in four smooth gauges, i.e., the Laplacian Abelian, maximally Abelian Wilson loop, and L-type gauges in comparison with the maximally Abelian ͑MA͒ gauge. They all reproduce the string tension in good agreement with the SU(2) string tension. The MA gauge is not the only choice of a good gauge which is suitable for the color confinement mechanism. Using an inverse Monte Carlo method and block spin transformation, we determine the effective monopole actions and the renormalization group ͑RG͒ flows of its coupling constants in various Abelian projection schemes. Every RG flow appears to converge to a unique curve which suggests gauge independence in the infrared region.