| I have created a resource that I think addresses your dissatisfaction. The information is available on physics.stackexchange
https://physics.stackexchange.com/a/670705 I use 'Hamilton's stationary action' to refer to the action concept of Classical Mechanics. For Hamilton's stationary action the standard presentation is that it is demonstrated that F=ma can be recovered from Hamilton's stationary action. Here's the thing: in physics it is common that derivation can be performed in either direction, and that applies in this case too. Hamilton's stationary action can be derived from F=ma The derivation proceeds in two stages:
1. Derivation of the Work-Energy theorem from F=ma
2. Demonstration that in all cases where the Work-Energy theorem holds good Hamilton's stationary action will hold good also Importantly, it's not retracing of the steps. The from-F=ma-to-Hamilton derivation hinges on the Work-Energy theorem. It's a different path altogether. The steps of the derivation show why Hamilton's stationary action holds good. It achieves the justification you are looking for. The derivation that I present is for the case of Hamilton's stationary action specifically; I'm positive the reasoning generalizes to all areas where an action concept is applied. The demonstration is illustrated with interactive diagrams.
(On physics.stackexchange the diagrams are posted as animated GIFs, the frames of the GIF are successive screenshots of the interactive diagram.) Each diagram has one or more sliders, to explore variation of a trial trajectory. The diagram shows how the kinetic energy and potential energy respond to variation sweep. The interactive diagrams are on my own website:
http://cleonis.nl/physics/phys256/energy_position_equation.p... |