The Elastic Problem is formulated for systems of rigid bodies equipped with both elastic organs and constraints. Lagrangian quantities are first introduced, as: (i) Lagrangian parameters, i.e., free displacements identifying the current position of the skeleton system (i.e., the system deprived of elastic organs), and (ii) Lagrangian forces, i.e., forces expending work over the Lagrangian parameters. A strain analysis is carried out, aimed to express the strains as function of the Lagrangian parameters only, and accounting for constraint prescribed displacements. The possible existence of floppy modes is discussed. Then, a stress analysis is performed. It is aimed at building up Lagrangian equilibrium equations, in which the reactive forces are filtered from the original balance equations. Some duality properties, similar to those holding for unconstrained systems, are proved. They lead to formulate the Virtual Work Principle (VWP), which, in its extended form, equates the external active plus reactive virtual works to the internal virtual work. The two corollaries of the VWP are also discussed and their use illustrated for: (i) directly filtering the reactive forces in the balance equation, or (ii) for obtaining kinematic compatibility conditions without manipulating the congruence equations (Complementary Virtual Work Principle). The VWP is successively used for constructing the Generalized Displacement Formula, a tool useful to compute displacement components when strains and displacements at constraint are assigned. The Constrained Elastic Problem is finally formulated and existence and uniqueness of the solution briefly commented. The problem is solved either via Direct Formulation, in which all the basic equations are tackled, or via Virtual Work or Energy Formulations, in which part of equations are replaced by a variational equation. Moreover, the variational approach is used to directly obtain Lagrangian quantities.