The video didn't make this clear, but it is answering real questions. Its used in the study of geometrical optics, celestial navigation, etc. Ie set up a 'toy' problem that gives you insights into rather more complicated dynamical systems.

From Tabachnikov's book:

Classical mechanics and geometrical optics, discussed in this chapter, are intimately related. The configuration trajectories of mechanical systems are extremals of a variational principle, similar to the trajectories of light. In fact, mechanics can be described as akind of geometrical optics; this was Hamilton’s approach to mechanics(see [3] for details). The brachistochrone problem is a good example of this optics-mechanics analogy.