Hamilton believed that a principle of least action applied to the whole universe must be rejected and replaced by a principle of stationary action.

He made his contention from his investigations in geometric optics and rediscovered Fermat’s principle of least time for the law of refraction. He found that light going from one space-time point to another always chooses the path with the least time which agreed with the idea that velocity of light is inversely proportional to the index of refraction. However, in space-time of extreme curvature where the end points coincide at equal time then the path is separated into two disjointed closed paths of equal radii which are topologically equivalent to the paths traverse along a 360° twist Möbius band. Note that ordinary Möbius band is simply 180° twist forming a one-sided 2D loop. Its irreducible 1D topology forms 2 disjointed loops with one twice the radius of the other.