In the previous chapter, various variational principles of differential type and their corresponding equations were described. When dealing with a constrained mechanical system, an appropriate method can be chosen from these approaches to solve the underlying problem more effectively. This chapter discusses several applications of these methods. The stability of the motion of a dumb-bell-shaped satellite is analyzed using Lagrange’s equation, demonstrating the effect of the gravity-gradient torque in a central gravitational field. The small oscillations of a conservative system are examined within the Lagrange framework, which has many applications in vibration analysis. A useful method called Routhian reduction is introduced for Lagrangian systems possessing cyclic coordinates. This method is then applied to study the dynamics of a heavy top, allowing for a deeper analysis of the motion. The Routhian dissipation function is also affiliated with the Lagrangian to accommodate dissipative systems. Both the Lagrange method and the Appell–Kane method are used to establish the equations of motion for the rolling of two wheels connected by a rigid rod moving on an inclined plane. The Kirchhoff equation is derived, which can be used to establish equations of motion for a vehicle with the Lagrangian expressed in terms of quasi-velocities, such as velocity in the body frame or angular velocity. Not only mechanical systems but also lattice vibrations in atomic physics and the motion of a charged particle in an electromagnetic field can be treated within the Lagrange framework, as shown in this chapter.