Understanding fish-like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanism. In this ar- ticle, we study a coupled system of partial differential equations and ordinary differential equations which models the motion of self-propelled deformable bodies (called swimmers) in an potential fluid flow. The deformations being prescribed, we apply the least action principle of Lagrangian mechanics to de- termine the equations of the inferred motion. We prove that the swimmers degrees of freedom solve a second order system of nonlinear ordinary differen- tial equations. Under suitable smoothness assumptions on the fluid's domain boundary and on the given deformations, we prove the existence and regularity of the bodies rigid motions, up to a collision between two swimmers or between a swimmer with the boundary of the fluid. Then we compute explicitly the Euler-Lagrange equations in terms of the geometric data of the bodies and of the value of the fluid's harmonic potential on the boundary of the fluid.