The dynamics of double-proton transfer reaction in formic acid dimer is investigated by performing ab initio molecular dynamics simulations. From the viewpoint of optimized energetics alone, the synchronous (simultaneous) proton transfer is more favorable than the successive one. However, a full-dimensional classical dynamics shows that there is a certain time lag, about 8 fs in average, between two proton transfers. When a proton undergoes the first transfer, it moves from an oxygen with higher electron density to the counterpart having the lower one. The proton thus needs an energy sufficient enough to break the chemical bond, resulting in a clime of a potential barrier. On the other hand, the second proton moves from the lower electron-density oxygen atom to the higher one. Hence, the second proton is shifted predominantly by the thus-formed electronic field. Not only due to the time lag observed but mainly because of the difference in the mechanism of transfer, therefore, the present double-proton transfer is identified as successive. A detailed study on dynamics shows that the vibrational modes of the O-C-O skeletons dominate the second proton transfer.