There is a way of _arriving_ at that subtraction, rather than just throwing it out there.

A resource I created:

Calculus of Variations as applied in physics: http://cleonis.nl/physics/phys256/calculus_variations.php

Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...

In that resource I show why it works.

In an earlier answer I gave more information about that resource. To find that earlier answer: go up to the entire thread, and search on the page for my nick: Cleonis