In the 1980s, Lee proposed an energy-preserving discrete mechanics with variable time steps by taking time (discrete) as a dynamical variable [Lee82,Lee87]. On the other hand, motivated by the symplectic property of Lagrangian mechanics, a version of discrete Lagrangian mechanics has been developed and variational integrators that preserve discrete symplectic 2-form have been obtained [MPS98,MV91,Ves88,Ves91a,WM97], but variational integrators obtained in this way fix the time steps and consequently, they are not energy-preserving in general.