Rongqing Dai
Abstract
For the past hundred plus years, modern physicists have been building the edifice of theoretical physics with complex mathematical representations of “action”, but have never satisfactorily answered the question what the action really is.
By bringing “action”, the fundamental building block in the modern physics, back to the classical context, we could acquire a better understanding of the essence of “action” and the corresponding principle. The purpose of doing so is mainly to help to answer what the action really is, which the academia of physics has failed to do for the past centuries..
Keywords: Action, Force, Time, Distance, Energy
1. Introduction
The mathematical framework of modern physics is constructed with “action” which is a concept established by classical physicists in the 18th century when modern physics had not yet sprouted. Before the era of modern physics, Newtonian mechanics was dominating the classical physics; however, after physics entered the quantum age in the 20th century, because of the theory of wave-particle duality and the Copenhagen interpretation, physicists no longer apply Newtonian mechanics to the motion of microscopic particles; instead, Lagrangian mechanics and Hamiltonian mechanics, which are based on the principle of least “action” and are considered equivalent to Newtonian mechanics in the classical context, were chosen to be the mathematical framework for describing the state of motion in the quantum domain.
But the problem is that never has the exact physical meaning of “action” or what exactly it is been clearly explained by anyone, whether in classical physics or in modern physics. Academics in physics just tell the world that the action is kinetic energy minus potential energy integrated over time, or the integral of the Lagrangian function with respect to time. . . . Sometimes it is also said that “action” can be the integral of momentum over distance. . . . Physicists are simply unable to give “action” a clear explanation in the same way that Newtonian mechanics gives “force” which can be easily related to real life experience by each of us. Some scholars say that the reason to use kinetic energy minus potential energy is to reflect the balance between kinetic energy and potential energy, but they cannot explain why it is the product of energy and time rather than the product of energy and any other physical quantity. The catch here is that no one can explain clearly what the physical meaning of the product of energy and time is. That is to say, the modern physics is largely based on a concept whose exact physical meaning no one has been able to clearly explain so far.
In fact, the establishment and application of the concept of “action” as well as the failure of a clear explanation for it over the history has revealed to us such a dim picture of the scientific status of “action”: it is only a mathematical pattern discovered by physicists of 18th century after immersing themselves in the mathematical play of mechanical problems for a long time, with no one able to explain exactly what this mathematical pattern means up to nowadays quantum era.
Despite the much heavier dependence on “action” in quantum mechanics, we might find that it would be indeed easier for us to make a better sense of its physical meaning in the classical domain:
According to Newtonian mechanics, there is a conversion relationship between energy and the work done by the force, and a conversion relationship between momentum and the impulse produced by the force; the work done by a force is equal to force times distance, and the impulse produced by a force is equal to force times time. Therefore, both energy times time and momentum times distance can actually be viewed as the result of force times time and distance. That is to say, in a certain sense, we can regard the so-called “action” as a reflection of the product of force and time and distance in the classical context.
2. The Essence of Action in Classical Mechanics
Now let’s suppose T0 to be the initial kinetic energy and V0 to be the initial potential energy, and L0 to be the initial difference between T0 and V0:
L0 = T0 – V0 (1)
Let’s also suppose ∆E to be the amount of energy transferred from potential energy to kinetic energy from time t0 to time t, and then at time t the integrand of “action” would be:
L = T – V = L0 + 2∆E (2)
If during this time interval the potential energy increases (or the object tries to escape from the potential field), then ∆E is negative, and if during this time interval the potential energy decreases (or the object moves towards source of the potential field), then ∆E is positive, and if during this time interval the potential energy changes not, then ∆E is zero.
Suppose during the small time interval ∆t, the incremental action is ∆S, the displacement of the object (mass point) is ∆r, the potential (energy) force is F, then we have:
∆S = L0∆t + 2 ∆E∆t = L0∆t + 2 F∆r∆t = (L0 + 2F∆r)∆t (3)
Therefore, the action S would be:
S = ∫∆S = ∫(L0 + 2 F∆r)∆t (4)
The value of the kinetic energy would change from one frame of reference to another, and we can always hypothetically choose our frame of reference with a constant velocity of the same as the initial velocity of the object (mass point) in question so that the initial kinetic energy T0 would be nil, then we would have:
S = ∫∆S = ∫(– V0 + 2 F∆r)∆t (5)
where V0 could be envisaged as the work of a preexist virtual force P by slowly (without acceleration) pulling the object from its end position to the initial position of that object in the motion, i.e.
V0 = ∫P∆r (5a)
So
S = ∫– P ∆r ∆t +∫ 2 F∆r ∆t (6)
From equation (6) we can have following conclusion about the “action”:
In classical mechanics, the essence of the (seemingly mysterious) action is indeed the integral of force over distance (displacement) and time.
With this force-based view of “action”, we might draw the following conclusion concerning the essence of the principle of least action in the classical mechanics:
In nature, the integral effect of some virtual force over time and distance plus the integral effect of the potential field force over time and distance would tend to be minimal.
In general, the two integrations in (6) could take different paths. But since P is a virtual force, we can select its path the same as the real path of F, then (6) would become:
S = ∫(–P∆t + 2 F∆t) ∆r (6a)
The reason why we still need to leave ∆t in the integrand is because we have previously assumed that the pulling of P was slow without acceleration, and in that case theoretically it will take the virtual force P an infinitively long time to pull the object from its end position to the initial position of the motion, while ∆t for the real force F would be real limited duration.
However, once again, since ∫–P∆t∆r is the virtual work done by some virtual force, we might assume that the direction of P varies so that it first pull the object to accelerate and then push the object to decelerate in order to assure that the object would have zero velocity at the beginning and at the end of the pull. By doing that we might take ∆t in (6a) out of the parentheses and get:
S = ∫(–P + 2 F) ∆r∆t (6b)
Here an obvious advantage of viewing “action” as the sum of the products of force and time and distance is that it can provide us with a more intuitive sense of “action” than simply energy multiplied by time or momentum multiplied by distance, since force, time and distance are the most fundamental dynamic elements (physical quantities) in everyday life. After all, in classical mechanics force is the form of the interaction between objects while time and distance are the ranges of interactions between objects – this might be why the 18th century physicists called the integration of energy over time or momentum over distance an “action”.
The fact that P in (6b) is a virtual force would make the sum –P + 2 F also a virtual force despite that F is the real force. With this force-based view of “action”, we might draw the following conclusion concerning the essence of the principle of least action in the classical mechanics:
In nature, the integral effect of some virtual force over time and distance would tend to be minimal.
Or
In nature, the sum of the product of some virtual force with time and distance would tend to be minimal.
For the sake of brevity, we might name the above expressions as force-based expression of the principle of least action.
In case when F is pointing toa center of potential (e.g. in the gravitational field), suppose F is independent of velocity, that is to say, F is a function of position only, F = F(r), since P is also a virtual force, we might also take it as a function of position only, P = P (r). With these preconditions, given that the initial speed is zero, we will have a straight line motion for the object.
2.1. The principle of stationary action
Considering that action S in physics sometimes take the maximum values instead of minimum values, the principle of least action has been often called as principle of stationary action [[1]]; accordingly, we might extend the above force-based expression of the principle of least action into a more general form as:
In nature, the integral effect of some virtual force over time and distance would tend to take extreme values.
Or
In nature, the sum of the product of some virtual force with time and distance would tend take extreme values.
And we might name the above expressions as force-based expression of the principle of stationary action.
Now by bringing “action”, the fundamental building block in the modern physics, back to the classical context, we could acquire a better understanding of the essence of “action” and the corresponding principle. The purpose of doing so is mainly to help to answer what the action really is, which the academia of physics has failed to do for the past centuries.
3. Discussion
One problem that can be easily spotted from (6b) is that it does not contain information about velocity, while the well known action-based Lagrangian and Hamiltonian mechanics can be used to solve the full ensemble details of the motion. We might find out the reason behind this from equation (2) and (3) when we reduce the dynamic L to the constant initial value L0 and the energy transfer ∆E, and then turn ∆E into the work of the potential (field) force F. By doing so, we lose the information of velocity of the motion. But on the other hand, we gain the benefit of seeing the physical essence of the seemingly mysterious ancient “action” that the academia of physics has failed to explain to us.
Consequently, the force-based view of action and the force-based expression of the principle of least action are not advantageous for mathematically solving the dynamics of motion but advantageous for comprehending the essence of the principle of least action. This better comprehension of the essence of the principle of least action itself is very meaningful for the future development of physics given that principle of least action is part of the theoretical foundation of quantum mechanics.
4. Final Remarks
The above presented force-based view of “action” seems to be missing in any physics textbook or literature, either classical or modern. While the seldom access to the aged literature from long ago by nowadays readers (e.g. the author of this writing) might account for why the said view appears to be missed in the classical literature, the missing of that kind of view in the modern literature is not a surprise – because the modern literature would not explain “action” in this way at all.
The reason why the modern physics does not interpret “action” in terms of force is actually very simple: in modern physics, force is no longer the most fundamental element as it was in the classical physics. The general theory of relativity says “gravity is not a force”; in quantum field theory, energy can pop out of empty vacuum and then disappear without any work done by any direct force. Modern physicists have strived to make energy and momentum as the most fundamental elements of nature, thereby depriving force of its title of the fundamental element of physics. Accordingly, it is naturally impossible for the modern physics to use the product of force with time and distance to explain the meaning of “action”. But unfortunately, with energy and momentum modern physicists could only give different mathematical definitions of “action” in different circumstances without a clear account for the essence of “action” in general.
Consequently, for the past hundred plus years, modern physicists have been building the edifice of theoretical physics with complex mathematical representations of “action” as its fundamental notion, but have never satisfactorily answered the question what the action really is.
[[1]] Wikipedia. “Action (physics)”. Retrieved from: https://en.wikipedia.org/wiki/Action_(physics). last edited on 31 January 2024, at 00:09 (UTC).
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