Micro-electro-mechanical systems (MEMSs) often use beam or plate shaped conductors that can be very thin—with (in terms of the thickness and length of the beam or side of a square plate). Such MEMS devices find applications in microsensors, micro-actuators, microjets, microspeakers, and other systems where the conducting beams or plates oscillate at very high frequencies. Conventional boundary element method analysis of the electric field in a region exterior to such thin conductors can become difficult to carry out accurately and efficiently—especially since MEMS analysis requires computation of charge densities (and then surface traction) separately on the top and bottom surfaces of such beams. A new boundary integral equation has been proposed to handle the computation of charge densities for such high aspect ratio geometries. In the current work, this has been coupled with the finite element method to obtain the response behavior of devices made of such high aspect ratio structural members. This coupling of electrical and mechanical problems is carried out using a Newton scheme based on a Lagrangian description of the electrical and mechanical domains. The numerical results are presented in this paper for the dynamic behavior of the coupled MEMS without damping. The effect of gap between a beam and the ground, on mechanical response of a beam subjected to increasing electric potential, is studied carefully. Damping is considered in the companion paper (Ghosh and Mukherjee, 2009, “Fully Lagrangian Modeling of Dynamics of MEMS With Thin Beams—Part II: Damped Vibrations,” ASME J. Appl. Mech. 76, p. 051008).