What are the key differences between the Lagrangian and Hamiltonian formulations of classical mechanics?

The key difference between the Lagrangian and Hamiltonian formulations of classical mechanics lies in the mathematical approach used to describe the motion of a system. In the Lagrangian formulation, the system's motion is described using generalized coordinates and velocities, while in the Hamiltonian formulation, the system's motion is described using generalized coordinates and momenta. Both formulations are equivalent and can be used to derive the equations of motion for a system, but they offer different perspectives on the system's dynamics.