What I see is just the covering.
The most important is invisible…
A. de Saint Exupery
Aristotle once said “Character is revealed by action” and Plank “Action is discrete”.
As one goes through more advanced expositions of physics, the concept of action is gradually given more importance, with it being considered a fundamental piece in some introductions to Quantum Field Theory through the use of the “path integral approach”.
The basic idea behind using the action is to assign a number to each possible state of a system. The function that does so is named the Lagrangian function, and it encodes the physics of the system (i.e. how do different parts of the system affect each other; i.e. holism).
Richard Feynman’s “least-action” approach shows that it is just classical physics constrained by a simple mechanism. When the complicated mathematics is left aside, valuable insights are gained.
So when we think of planets orbiting the sun, we usually adopt Newton’s view that they are constantly accelerating -in this case changing direction- in response to gravitational forces. From this, we can calculate the motions precisely, and the impressive accuracy of predictions for total solar eclipses shows how well it works.
There is, however, another way of thinking about what is happening that gives exactly the same results. Instead of the Principle of Acceleration by Forces, as we might call it, there is an alternative called the Principle of Least Action, or more correctly, Hamilton’s Principle. It is a principle that was first put forward about fifty years after Newton’s, in its earliest form by the Frenchman Pierre Maupertuis, and in its ultimate form by the Irishman William Rowan Hamilton.
The general idea is that when a planet travels through space, or a ball travels through the air, the path that is followed is the one that minimizes something called the action between the start and end points. Action is just something that can be measured out for some particular object moving along a particular path. It is exactly defined and is measured in units of energy multiplied by time. The details are not important unless someone has the need to make calculations. We therefore have two quite different ways of describing situations in classical physics that are equally good in terms of giving the right answer. To give the simplest possible example, we can think of a golf ball travelling across an idealized, frictionless, flat green. In Newton’s view, the ball moves in a straight line at constant speed, because that is what Newton’s Law says it must do. In Maupertuis’ view, the ball does this because this path is the one that has the least action between the start and end points. This trivial example can be made more interesting by making the green have humps and dips, which are like having forces acting on the ball, but the principles stay the same.
Hamilton’s Principle is fundamentally equivalent to Newton’s Laws, and comes into its own when solving more advanced types of classical problems. But as an explanation, it means that things need to know where they are going before they work out how to get there.
Actually, this is where classical mechanics makes its first big step toward quantum mechanics, if only we look at it another way. The mathematics of Hamilton’s Principle can be described in words alternatively like this: given its starting points and motion, an object will end up at locations that are connected to its starting point by a path whose action is a minimum compared to neighboring paths. If locations away from the classical path are considered, no such paths exist-there will always be a path with the least action, but this is not a minimum.
It is an unfamiliar idea, but well worth a little effort to try and digest. One vital change to note is that the emphasis has moved away from knowing the path that is followed to having a test to check whether possible destinations are on the right track. And the crucial factor is being able to compare the actions of different paths.
If we stay within the world of classical physics, we can choose to ignore this strange new description and stick with the more comfortable idea that things are accelerated along paths by forces, but this would be a personal preference rather than a rational one. The new view prompts the question: “How do things work out whether possible destinations are linked to the start by a path of minimal action?” We should appreciate, however, that the old Newtonian view prompts equally difficult questions like: “How do things respond to forces by accelerating just the required amount, instant by instant?” Moreover, as we all know, the “action version” is the one that the world around us seems to use, and that’s when Prometheus Project comes in taking the action question seriously and giving it a rather simple answer: Nature has to check out all possible destinations to see if they are on the right track. It must do this by trying to find out if there is a path of “minimal action” to each destination; an answer that brings of course another question: Why do people are stuck with action and don’t know that the principle of minimal action really works? (Wondering Logically is Science)
The answer is simply because they don’t know that it has to be constructed to do so.
Credit for the formulation of the principle of minimum action is commonly given to Pierre Louis Maupertuis, who felt that “Nature is thrifty in all its actions”, and applied the principle broadly.
So, we intentionally look for functions that, being used in integral along a path from start to end state, give a function of path which indicates realizable paths by being minimal in action. When we’re lucky to find such function this method works. And Hamilton’s Principle just says we should be lucky in this regard.
