Weak formulations in Analytical Dynamics are developed, paralleling the variational methods in elastostatics, and including a fundamental yet novel approach for treating constraints (both holonomic and nonholonomic). A general three-field approach is presented, in which the momentum balance conditions, the compatibility conditions between displacement and velocity, the constitutive relations and the displacement and momentum boundary conditions are all enforced in weak form. A primal, or kinematic formulation is developed from the general form by enforcing the compatibility conditions and displacement boundary conditions a priori. The conditional stability of the kinematic formulation is the counterpart of the locking phenomenon in elastostatics and may be avoided, either by reduced order integration, or by utilizing a mixed formulation. Toward this end, a two-field mixed formulation is presented, which follows from the general form, when the constitutive relations are satisfieda priori. A general set of the constraint equations is introduced into the kinematic and mixed formulations, using a specific choice of multipliers, which results in modified variational principles. Several simple examples concerning rigid body dynamics are presented.