Our study addresses the first boundary value problem in the context of the theory of elastic Cosserat bodies, in its classical form, that is: the motion equations, the initial conditions, and the boundary relations. Then, the variational form is attached to this problem. In this context, the elasticity operator is defined and its properties are proven, the most important of which is that this operator is positive definite. Based on this last property it is possible to prove a result of the existence of a solution for the formulated boundary value problem. Also based on the positive definition of the elasticity operator, is highlighted the possibility of approximating the solution using a variational method.