Just had a few Eureka moments.
All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.
First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.
fixed: which changes only if estimated wrong.
That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics.
Least action means zero allowance beyond what you already have, no budgetary provisions for extra ability. Light does not have it, particles don’t you and I don’t have. But we certainly do have abilities, just that its limited.
So very quickly the “Inherent ability = difficulty * accomplishment” intuitive law turns into a physical principle for light. Light will go through straight line, but which one? action says difficulty is same in all directions so it will go in all directions and it does so with equal accomplishment. It covers same distance in all directions.
But what if difficulty changes in a particular direction? It can only increase from the minimum. So accomplishment will be less. Hence a particular path is the only possibility where accomplishment * difficulty is again same as what it was when difficulty was less but accomplishment was more. So now in the more difficulty scenario accomplishment will be less. Less amount of path will be covered –like swamp and sand case.
This is exactly what the Fermat’s principle of least time is. Its actually lazy to call it least time, more general is least- optical path — adjustment of path wrt how much difficulty is faced in moving that path and still more general is least- action. Some people take objection at least saying it should be stationary but if you see that least means zero variation its same as stationary. There is nothing to be alarmed here, one should follow the maths correctly than be worried too much about naming.
So now “Inherent ability = difficulty * accomplishment” becomes in Physics “action = momentum * distance”. Remember that its the same as “action = energy * time”.
- Ball goes distance x
- Lets say a ball goes a distance of x at a given momentum p, the ball has an inherent ability or action of px.
- Ball goes distance 2x
- Now the ball goes a distance of 2x at the same momentum p, the ball has an inherent ability or action of 2px. Evidently in 2nd case the ball has twice as much inherent ability or action.
- Crew works for time t
- If your crew works for a time t at a given energy E, the crew has an action of Et.
- Crew works for time 2t
- Now he/she works for a time 2t at the same energy E, the crew has an action of 2Et. Evidently the crew has twice as much inherent ability or action.
Its this action which is least, stationary, limited or allowed to change a zilch, but thats all. Just because you spent it unwisely does not mean nature will provide more.
While we have been arguing time, path or optical path they are just specific validity. In general its the action which governs our mechanistic decisions. We are choice-less because we do not have any inherent ability for those choices, in other words we do not have sufficient action.
Remember that its technically really sound, its not just brain-work. You can for yourself see the relationship between all variables talked about here and they are all consistent.