The why is actually quite simple- and the same reason the Fourier transform is sometimes used. Some problems are simply more elegantly expressed in a particular

The why is actually quite simple- and the same reason the Fourier transform is sometimes used. Some problems are simply more elegantly expressed in a particular basis system. Nobody tries to express a ball's motion in flight via Fourier analysis, but you certainly could. In the same way, sometimes solutions for the position(s) of a system is the most natural basis system for describing/investigating a problem (use the Lagrangian) and sometimes solutions for its momentum are (use the Hamiltonian). Of course, they're intrinsically linked since the evolution of one determines the evolution of the other.