With the recent progress of MEMS technology, the development of MEMS devices for radio frequency (RF) applications has been growing rapidly. RF MEMS devices have a broad range of potential applications in wireless communication, space, instrumentation, etc. The main failure analysis issue for RF MEMS metal contacting switches focuses on contact metallurgy. During passage of current through the switch contact, some portions of the interface may melt and weld due to high temperature and a molten metal might be splashed from the surfaces, which leads to device failure. In this study, a mathematical treatment of the general problem of electrical contacts and heating related to MEMS switch contacts is developed. The spatial distribution of the potential, the current and the temperature in a special case of electrical contact between two gold bodies whose thermal and electrical conductivities vary with temperature is analyzed, and an explanation for the collapse of gold contact system before reaching the melting point is given. For this purpose three different approaches are applied: 1) exact solution, 2) simple model approximation, and 3) variational calculus.