In this paper we study a coupled system of partial differential equations and ordinary differential equations. This system is a model for the 3d interactive free motion of rigid bodies immersed in an ideal fluid. Applying the least action principle of Lagrangian mechanics, we prove that the solids degrees of freedom, solve a second order system of nonlinear ordinary differential equations. Under suitable smoothness assumptions on the solids and on the fluid's domain boundaries, we prove the existence and $C^\infty$ regularity, up to a collision between solids or between a solid with the boundary of the fluid domain, of solids motion. The case of an infinite cylinder surrounded by a fluid occupying an half space, we prove that collisions with non zero relative velocity can occur.