To address constrained problems, the unknown constraint forces and/or torques must be included in Newton’s equations and/or Euler’s equations, resulting in additional variables that have to be solved. Alternatively, the observation that the constraint forces do no virtual work for virtual displacements compatible with the constraints leads to the variational approach in analytical mechanics. Various differential variational principles of mechanics are described in this chapter, including the d’Alembert Principle, the Gauss Principle of least constraint, and the Principle of Virtual Power. These corresponding variational equations are then used to derive the equations of motion without including the constraint forces. In particular, the method using Lagrange’s equations, characterized by a Lagrangian function of the generalized coordinates, leads to an analysis in Lagrangian mechanics. The Gibbs–Appell equation uses the Gibbs function of accelerations to establish the equations of motion in terms of the generalized coordinates or quasi-coordinates. Based on the Principle of Virtual Power, the Jourdain variational equation is considered more intrinsic in dealing with velocity constraints. The Appell–Kane equation can be used to derive the minimum number of equations needed to be solved. Finally, the Principle of Virtual Power is extended in this chapter to treat the impulsive motions subject to impulsive constraints.