The principle of least action is that the integral of the Lagrangian is minimized - but, as this is intimately connected with total energy in most investigations.

Exactly. But what does this

**mean** ??

Could you expand on what you mean by "intimately connected" ?

To answer this question, can you describe a universe (with conserved energy) in which least action is not abided by? What sorts of differences would we see with falling apples and trajectories of objects and such? (...or use any other toy example that best describes these differences)

Now those questions can be answered in several different ways. One way to answer them would be saying,

*"If energy is conserved, it must be the case that the integral over the Lagrangian is minimized".* i.e. PoLA is a different way of phrasing energy conservation. If one, then the other. If energy conserved, then PoLA falls out at the bottom of the proverbial chalkboard.

Is that true? Or rather is PoLa a separable, disconnected property added on top or besides energy conservation? Or are they two ways of stating the same thing?

Again, the best way to answer these questions is describe some toy scenario in which energy is conserved, but PoLA is not abided by.

My best understanding at this point (today) is that the larger potential energy becomes, the faster it is converted back into kinetic energy, and the lower it becomes the slower it is converted. I'm looking for some sort of rule-of-thumb way of understanding PoLA as well as Hamilton's principle. I know exactly what the equations say and I can repeat them back to you on a test. I can blindly apply Hamilton's principles to exercises in a textbook with pinpoint accuracy and get my A+ in the class. But I want something more. I want a more well-rounded understanding of what this

**means** for our universe.