More on the Principle of Least Action

   It appears that the way the variation is done affects the minimum of the action integral. If one arranges the variation so that it is normal to tangent of the curve the number of terms contributing to the action integral is reduced to two.

Here's an exaggerated variation to make it more visible.

The variation in the segments contributing to the variation of the action integral show a fairly uniform variation with positive signs.

The resulting action integral now has a minimum when the magnitude of the variation is zero and we can rightfully refer to it as Least Action.