Some musings regarding the "Principle of Least Action"- not solely as that term is understood in physics today, but mainly as it is appropriate to other processes in the universe which exhibit minimizing characteristics. Written as a part of a dialogue with a friend, a few portions have been redacted here.
...Given any finite set of physical parameters (such as space, bodies or masses, different forces, etc.) in a system, there is always a particular state of the system which minimizes the potentials for further change. Whatever the current state of the physical system in question, the variations which take place in that system will always tend towards this state. Given the set of elementary forces and molecules of particular structure which exist in a system of low energy, the crystallization geometry observed represents the configuration in which the potential for further variations in the system are minimized- a stable geometry is produced and sustained. Akin to this kind of process is the way in which the economic organization of humanity would be “forced” into a new stable configuration -a supranational configuration beyond mere nation-state based economic planning- by the historical “forces” which are characterized by the initiatives which are intended to foster mankind’s mastery over the processes of the solar system, starting with Helium-3 mining and asteroid defense. We see then that the same kind of reasoning which is applied to physical systems like crystals in fact applies to all physical systems, human societies included.
Reminiscent of the second law of thermodynamics...I have spent considerable time thinking about this issue and, based on my judgement, I consider the second law of thermodynamics to be absolutely valid, but only for closed systems, that is, conceptually abstracted physical systems of a finite number of parameters otherwise known as thought experiments (which could never exist in reality). But, based on our LaRoucheian ontological framework, we see most definitely that the second law is not valid for the real physics of our universe. It could be argued that the universe is necessarily a closed system and thus the 2nd law is applicable with pure validity to the universe as a whole, even if only approximately applicable to semi-closed physical systems. I reject this argument because I do not accept the idea that the universe is a closed system.
Applications to human society
All this brings to mind the way in which the activity of mankind changes the universe, but with particular emphasis on the case of mankind's actions on himself. Take the process of political organizing for example: We exist in a society in which the political leadership and population adhere to a particular mental construct, which is shaped by a set of axioms which determines the behaviors of these people in any given context. These unfortunate people are doomed (without us) to bring about their own destruction because of the nature of this "cultural force", whose final "least-action" state is, unfortunately, unlike the aesthetically appealing character of crystalline least action states of molecular systems , a thermonuclear holocaust. Our political leaders will, like mindless molecules following their lines of force, follow their own least action paths, or minimal tension pathways, which lead, inevitably, to this horror. This final state, towards which civilization would plunge if the physical (cultural etc.) parameters of civilization were not changed, was forecast by many people utilizing this method (MacArthur, King, Kennedy, LaRouche et al.). The question arises then, how do we change this "trajectory"? Obviously, we need to induce a change in what you might call "the fundamental forces", or physical parameters, of the system. By inducing a change in this underlying geometry, we are enabled to accomplish what seemed impossible earlier. We can thus change the character of the final configuration towards which society is moving. Now, this considered, transfer your attention to the subject of mankind's capability to affect changes in the universe generally.
Applications to physical systems generally
Take the case of physical systems which are conceived of as primarily determined by the elementary forces of physics. Our conceptions of such systems imply that the capability to change the behavior of these systems is reliant on the manipulation of the forces which adhere inside of the system, which implies a capability to overcome the forces themselves. Our current concept of energy is defined as the capability to overcome a force. Thus, by successive increases in the capability to expend energy per unit time (as in the progression from wood, to coal, to oil, to nuclear to fusion, to matter/antimatter reactions) the ability to overcome greater forces over greater distances is achieved. But, what if we were to conceive of the forcible lines of action in the such a system as being nothing but the characteristics of the unique physical space-time within which the process is located? That is to say, given any set of forces and lines of force in a system, what if we were to construct a unique force-free physical space-time geometry which allows us to account for all of the observed effects in the system without making use of the concept of force at all? (for example- Einstein’s treatment of gravitational “force”) At first glance, it might not seem that this alternative way of conceiving of the physical system has any consequences with respect to our ability to change it, despite perhaps a factor of conceptual or calculational convenience in certain circumstances- After all, we would still require the same amount of energy to manipulate the physical system in the same way as we did before (when we conceived of forces as characterizing the systems behavior as opposed to force-free geometries). But, if we recall the way in which the culture -akin to a "physical geometry"- of human society can be changed such that the capability to accomplish certain things which previously seemed impossible then becomes possible- to change the directionality of the process itself, then we are taken by a sense of wonder when we consider how we might be able to affect changes in the underlying physical geometries of “abiotic” systems in order to accomplish those things previously thought impossible. What if we were able to induce a change in the very underlying physical space within which the process unfolds itself, thereby eliminating those very lines of force which we were previously trying so hard to overcome, and replace them with alternative lines of force which were conducive to our objective? Our attention is immediately called to the issue of controlled thermonuclear fusion, in which the primary obstacles which stand in our path seem to be the overwhelmingly extreme energetic parameters which must be met on the basis of a relatively brute-force energetic method of approach, but also, at the same time, we recall the way in which the phenomena of self organizing plasma structures provide us with the clues by which such limitations can be overcome as an example of the changes in physical geometry mentioned above.
With this in mind, we see that there is indeed an explicit limitation imposed on the capability of mankind to effect changes in the universe if we do not take this consideration into account and simply maintain the idea that mankind can only change the universe through the “brute-force” energy transfer methods which characterize the mastery and harnessing of various forms of “fuel sources”, as in the above referenced progression from wood to matter/antimatter for example. Antimatter fuel represents a limitation- there is no possibility of ever finding a fuel which contains more energy per unit mass than antimatter. Also, the rate at which such energy can be released is limited by the reaction speed, as well as the fuel delivery speed which itself is bounded by the speed of light. The capability for density increase is also bounded by the size of the antimatter particles which are utilized. Thus, we seem to come to the limit of the progressive rise of energy flux density of any fuel source which is conceivable. Are we to take this as defining the upper limit of mankind's capability to change the universe? No! The next phase, it seems, of mankind’s capacity to induce changes in the universe needs to be on a higher level that the relatively kinetic (even if refined) capabilities represented by the process of inducing changes solely through energy transfers. We now seem to come to the point at which the target of our activities are the very principles themselves which determine the energetic requirements associated with any particular change in a physical system. If we can change these within the processes of the universe generally, just as we do so already in human society, then our power is greatly increased. This calls directly to mind the identification by Vernadsky of the same problem: “Thought is not a form of energy, how then can it change material processes?”. We would be safe in identifying the principle of life in a similar way. Based on the available evidence respecting the power of life to change material processes on the planet more rapidly than any other known process (besides man), if we understand that life itself, like thought, is also not a form of energy, then we are led to believe that life itself represents a process by which the physical geometry of the universe is changed in order to bring about changes in the universe which are not capable of being induced through mechanically determined, or linear, energy transfers.This leads us to believe the following: A mastery of the higher order principles of universal life and cognition processes holds the key for the continued increase of mankind’s capability to change the universe- to increase his power over the universe- even beyond the class of capabilities currently conceived to be associated with the mastery of antimatter processes- all this in the fashion hypothesized above.
Elaboration of the general characteristics of the concept of change
Before proceeding onto the issue of the second, seemingly primary, way in which we can conceive of the PLA, let us refresh our memory as to what we are actually attempting to characterize with the PLA when we refer to “action” or “change”. Whenever we conceive of a change, no matter what kind of change it might be- whether it be elementary or complex, long or short, mental or physical or biological- three elements are always implied: The initial condition, the final condition, and the transformation by which the initial condition becomes the final condition. This considered, it is clear that when we we attempt to conceive of a reason as to why any particular change might occur, we seem to be unable to think of one simple reason which allows us to understand why the change occurs in the way it does as opposed to otherwise, because every time we try to do so we are confronted with these three differences which are implied by the very concept of change itself. Thus, we seem to need three different but related reasons to enable us to fully understand change: 1.)There must be a reason why that which is to be changed exists in the way it does. 2.)There must be a reason why the final state is such and not otherwise. 3.)There must be a reason why the transformation between the two occurs in the way it does and not otherwise. Stated again: 1.)That which is being changed, (or in other words: the initial condition) has a reason for its existence; 2.) That which it is changing into (or in other words- the goal, the objective, the intention) has a reason for its existence; 3.) The manner in which that which is changing becomes that which it is changing into has a reason as to why it is such and not otherwise. We see then that the concept of change itself is not simple, but that it has three necessary components which seem to require that we conceive of three different reasons for its existence in any circumstance.
Methods by which change is analyzed
We will notice that physics, under the influence of LaPlace before the quantum revolution, there was a unification of these reasons in a particular way- that is, towards an explanation of change in which the final state is nothing but the necessary result of any preceding state which changes according to a fixed rule. However, in the case of the PLA, this kind of reasoning is not applied and an explicit consideration of the final state is made in order to determine the third reason, that is, the reason as to why the system evolves in the way it does. In the case of the PLA, the way in which the system evolves is determined by the final state. This is clearly demonstrated in the case of the PLA’s application in standard physics. Planck writes: “Thus, by referring to a finite time integral [in order to determine the characteristics of the motion], the motion at any instant is investigated with the help of a latter motion, and present events are in a certain manner made dependent on later events, and the principle acquires a teleological character.” Planck seems to reject the idea that this teleological concept should be allowed to enter into the scientific picture of the world while at the same time emphasizing the importance of the PLA on the basis of it’s usefulness in calculations. For me, this calls to mind the issue which Lyn has raised respecting the debate over the methodological versus ontological transfinite- that is: Whether or not the higher-order geometrical/physical conceptions which are generated by the human mind for the purpose of providing a sufficient reason or cause as to why the world seems to behave the way it does-if these reasons- correspond to ontologically real characteristics of the universe; or, whether these concepts are simply useful abstractions which do not correspond to any reality outside of our conception of them- useful abstractions which simply provide us with a convenient way of making predictions and thinking about different things etc. Are we to consider the universe as functioning in a way in which the future determines action in the present because of the applicability of the PLA, or are we to consider the PLA merely a useful conceptual device which we can use to characterize physical changes? This question probably begs a clarification of the concept of “future” and of time itself- all somewhat mysterious concepts.
Lets see if we can elaborate how these two methods by which the change in a system is determined are different from each other. In one case, the case of mathematical equations of force and motion, we have a statement concerning the relationships between assumed quantities, which have maintained themselves in the past, through a process of measurement and analysis, and from this we arrived at general statements concerning the way in which these abstracted quantities relate to each other. From these general statements, or mathematical physical laws, we assume that the statement will always be true under any given condition, therefore, it is assumed, that if we are given any quantity or combinations of quantities which are present in any physical system, then the other quantities which relate to these quantities through the relationship in the mathematical law can be found. If one of the conditions in the mathematical law is time then we necessarily imply that if we are given a particular time then we can find the other conditions which are related to time and we call this predicting the “future”. But in this case, ultimately, the future is nothing but another form of the same predefined relationship which is established in our mathematical law. Even if there are certain quantities in the equation which themselves change as a function of the change in another quantity in the equation, then the future can still be determined if the equation for this additional change is also defined. Therefore, the mathematical laws of physics are said to rely on deductive logic, and in this sense, represent the rules by which a series of tautological statements can be made about the universe. However, the scientific validity of this approach relies upon the consistency of the relationship amongst the quantities in the equations. The evidence which is invoked to defend the idea that these consistencies will always be observed is non other that our past experience- We believe that if the mathematical relationships were never seen to deviate from the equations in the past, then we can safely assume that they will do so in the future. Therefore, the mathematical laws of physics rely also on inductive reasoning, the conclusions of which are incapable of being proven valid. Ultimately, the equations of mathematical physics are statements about the relationships in a system at every point in time- therefore, it is not so much that you are predicting the future, it is that you already establish the future as predefined. Every mathematical law which we conceive of as existing implies a particular static construction of the universe- all past present and future aspects of it. In this case, those “events in the future” could be conceived of as presently existing in a way which is no different that we conceive of an object in another room as presently existing despite the fact that they are not experienced now but could experience it later.
On the other hand, when we are dealing with the PLA, there is no such ultimate statement made about every spatial/temporal point in the system. Take the example from physics- the minimization of “S-action”, or the difference between the kinetic and potential energies over any time interval. It is only until you choose a final state that you are enabled to determine the necessary path which leads to it. However, what significance could this principle have if he future is defined in the way elaborated above- using the static picture of LaPlace?If you are still working within this framework, the PLA’s relevance is not quickly seen. If we are still selecting two points in pre-defined, static space/time and finding the connection with the PLA, then no challenge seems to be made to the world-view.
Implied significance of PLA inside and outside of standard physics
I have heard (although I would like to understand this more clearly) that the equations of classical mechanics can be derived from the PLA. Planck: “Now, the PLA furnishes, in every case, as many equations as there are variables.” Or from Dr. Stone: “You can actually throw away Newton’s laws and do all of classical mechanics just based on the PLA.” If my understanding of this is correct, it would seem to imply that the equations of mathematical physics lose the current arbitrariness which seems to plague them. For example, one well known indicator of the seemingly arbitrary nature of the equations of mathematical physics are the “constants”- Why are the constants the value that they are? Why not another value? It seems arbitrary. I would be interested to learn whether or not the PLA removes this particular arbitrariness by bestowing sufficient reason where there currently is none. One thing to restate about such equations however is that they are not statements about the “future”, they are statements about a static present, for in the case of a universe defined in such a way, the future is nothing but the word attributed to those things which are already defined as existing in a particular place (just in another dimension which we have limited access to), but which simply have not been experienced yet, just as we conceive of objects presently existing in another room for example. However, the PLA defines the present using the future. But there does seem to be a difference: the future states which the PLA can relate to, do not necessarily have to be future states which are already implicit under the currently existing “physical laws”. The two states in the process do not necessarily need to be the mere points in a static structure which is predefined by mathematical las. For example, JFK established the objective of putting a man on the moon. This future state of a man on the moon was not a future state which was implicit in the then existing “laws” of the universe; We did not have the science or technology to do it. Yet we could say that the PLA determined the organization of the economic process which led to that final objective being accomplished. Thus it seems that the PLA is applicable not just in the task of characterizing the connection between two different entities, such a points in space time which can be conceived of as statically existent at once, as is done in physics, but it seems that the PLA is applicable also to the connection of currently existing states to new existences yet to be made. Perhaps this is why it is so that when the PLA is applied to classical physics systems there is no capability to identify the effective difference between the two methods (using the PLA or classical mechanics). Perhaps then it is in the domain of qualitative-or nonlinear-change that the PLA actually most clearly shines forth and asserts it’s fundamental necessity. That is to say that since both the PLA and the mathematical laws of physics are applicable in the treatment of the kinds of changes investigated by classical physics, but only the PLA and not mathematical laws are applicable in the case of non-linear, qualitative, change. This question of qualitative change will be taken up again later. But before that it may be useful to examine some of the basic notions involved here.
Second way the PLA is conceived of
The second way of conceiving the PLA, which we have already spoken of, is the idea that, given any arbitrary objective, there is always a course of action by which that objective could be accomplished which minimizes the amount of effort applied. This is, of course, speaking in terms of human action. Could it perhaps be restated more generally to include any physical system?
Connection to he Principle of Necessary and Sufficient Reason
Before answering that I would just like to point out something: The PLA seems to me to be either identical to, or interrelated with the Principle of Necessary and Sufficient Reason. I have heard some people say that the PLA is derived from the PNSR but it is unclear to me what this might mean. Take this example:
Dan: "Hi Joe, I want to accomplish objective X. What is the best course of action to take which will enable me to do so?"
Joe: "Hi Dan. The best course of action for you to take is the course of action which is most reasonable."
Dan: "How do I know which course of action is the most reasonable?"
Joe: "It is the course of action which requires the least effort to accomplish your goal, the least action."
Dan: “How do I know what the path of least action is?”
Joe: “It is the path for which there is a reason to take it.”
We see here that the two principles seem to be one and the same. In the case of the motion of light too, these principles seem the same. (there could be no reason that light would move according to any other time path than the minimum path, because any other time value can be arrived at by an infinite number of different paths and thus no reason could be given as to light would travel on one such path as opposed to another).
What is “least action”
A question arises as to what exactly PLA -as a Universal principle- minimizes in a given physical change. For example, in the case of light, we say that, even though light seems to bend, it really just follows the path of least action, that is, the light follows the path which requires the least amount of time to move from one point in space to another point in space. In this case, the principle of least action expresses itself in the minimization of time taken to traverse a given length of space. But what about another example? Take the case of projectile motion. The sum of the difference between kinetic energy and potential energy over the path of the motion under the influence of forces will be minimized. This corresponds to the PLA as it is known in modern physics, not as a principle, but as a rule, since, in this case, “Action” is defined, ( the integral of energy differences over time- units of energy-seconds). Now, as broadly applicable as this least action rule may be to physics, it seems that we can conceive of it only as a specific way in which a more universal PLA manifests itself in those kinds of specific physical systems.
Lets look at another example in which our intuitive sense of a universal PLA is called to mind: Take the design of machinery. Two machines which produce the same amount of work in the same amount of time, and yet, one machine requires more electrical power. The least action preference in this case would be the machine which consumes less power. In this case the PLA seems to indicate that energy, or power is minimized.
Take another example. If I want to go to Seattle to discuss this matter with you in person I could either walk there, or I could fly there on a jet engine aircraft. Now, obviously I would fly there because It is certainly easier for me personally. In this case, would we not consider this to be the path of “least action”. However, we see something funny here, which is that this least action path I took on my journey did not minimize the “action” as defined in modern physics. In fact, the amount of energy consumed in this journey was far in excess of the amount which I would require to walk the whole way there because I need to use the thousands of gallons of jet fuel and haul a giant aircraft around and other things like that. So we see that, in another case in which the PLA prevails, yet another different physical parameter is minimized, namely human effort, or, roughly, man-hours, while another one- energy- is maximized. This seems to be true for most human economic processes generally. Improvements in manufacturing technology are always associated with a reduction of man-hours per unit work but with a corresponding increase in energy throughput, or energy flux density. .
So what does the PLA minimize!? Is it time? Is it energy-seconds? Is it energy? Is it man-hours? What is “Action”? Aye, there’s the rub. It seems as though the kind of change will determine the way in which the PLA comes to expresses itself.
The PLA is dependent upon quality of change
To get at this idea, lets look at light again. We conceive of light as moving from one physical location to another. According to the principle of least action, the light will always move such that the greatest amount of distance is traversed in the least amount of time. In this case, the PLA seems to express itself as a maximum, minimum function. The parameter of importance which is minimized, time, is what would represent the “action” in this case. However, what if we were to invert this relationship as follows: As opposed to the idea that given any two points in space light will traverse the path of least time, we say that given any two points in time the light will traverse a path of least distance. Now, in this case, we are indeed minimizing a certain parameter with respect to another. However, in the process of doing this, we lose the meaning of the kind of change which we are attempting to characterize in the first place, viz. motion of light. For we see that, given this inversion, there would be no motion at all. It seems evident that the quality- or perhaps we should say: the “agency”- of change which is being characterized determines the way in which the parameters relate to each other as well as implying the specific parameter which tends to be minimized according to the PLA. But why would this tendency towards minimization be necessary? Why not have some sort of other arbitrary rule by which the specific parameters relate to each other? Take the case of light again: Why not have a different rule by which the motion of light was governed? For example, why not have the rule that light always traverses the path between two points which is exactly one half of the least time, or maybe one third, or nine tenths of the least time? Wouldn't any of these be just as good of a law that light could follow? We see that this could not be the case, for if light were compelled to move on the path which took any other time than the path of of least time, then the light would not know which path to take, considering the fact that there are an infinite number of paths between any two given points which correspond to any time interval other than the interval of least time. Therefore, it would be impossible to know why the light chose to take one path rather than another one out of the infinite paths available. On the other hand, if the light always follows the path of least time, then there is only one single path which will be taken, which can be known. Therefore, we see that it is coherent with the Principle of Necessary and Sufficient Reason that the light moves in this “least action” fashion, as noted above.
Application to biology
Perhaps we can apply a similar line of reasoning to any physical process- that the change it takes is the change of least action and of necessary reason. In the case of biological selection of isomers for example and Vernadsky’s identification of the necessity of a factor which renders the different isomers not functionally identical, we see a case which begs similar questions.What is the principle of change which implies a particular selection of isomers in it’s least action characteristics? Why is it that this selection is the necessary selection as opposed to the other, just as the light’s path selection is necessary as opposed to the others?
Life seems to express a certain “efficiency” in its various expressions, such that, for any particular change in the universe which is produced by life, there is no other conceivable process which could have brought about the same change in a more efficient way. For example, the conversion of the light energy from the sun into chemical energy via photosynthesis- could any other method be conceived of which converts a given amount of radiation into chemical energy which was more efficient? Taking the idea to a larger scale, if we conceived of the sun as a conscious entity, could there be any other way more efficient conceived of, other than the way in which we observe how life acts, in which the sun, using its energy, could cause changes in the earth’s crust of the type we see generated by life today? We the PLA staring at us again. The kinds of changes which the universe itself strives to manifest are only possible of being brought about by living processes it seems. The same hold true even for individual organisms, indeed, each individual organism seems perfectly designed to carry out the purpose of producing its own particular effect on the universe- they are thus designed in the fashion implied by the PLA.
Human action and the PLA
Lets look at some examples of human activity to see if we can uncover anything more about the PLA.. If I want to get into the office in the morning I take the path of least time. However, in this case, I can conceive of alternative routes which I could take if I wanted to. In this sense, I am not like light, which literally could not move at all unless it moves in the path of least time. I can chose to go on any path I wish. Where is the necessity for minimization here? The issue perhaps lies in the notion of “goal” referenced above. There is no human objective which takes place in an abstract vacuum. That is, you can never only have one simple goal like “going to the office”, for, if that were your only goal, then you could never possibly choose which route to take to reach the office since it wouldn't matter to you that you get there in 5 minutes or 5 years or 5 million years. There are an infinite number of routes and transportation methods which you could take. There is no possible reason that could be given as to why one route should be chosen over any other in this case of an abstract isolated goal. However, if we add a “causal context”, or, in other words, a larger goal, within which the specific goal of reaching the office is contained as a necessary feature, then we achieve the capability to differentiate between the options before us in fulfilling the specific goal. For example, we have an objective to raise funds for our beautiful organization that day. This delimits a time interval which we can spend on transportation which delimits the routes which you can take in order to get to the office in time to have any time on the phones. However, this is still not very specific, because you could still take an excessively long route and get to the office just one minute before the time to close up shop and get only one minute of calls in. Knowledge is required for this, of how long of a time interval is required to raise money, which is also related to how much money you want to raise, etc. Not only that, but if you also want to do other things that day besides travel to the office and make calls, such as use the restroom, or eat, or sleep, or read the briefing, or watch a video on LaRouchePAC, then this will also factor into the limits of time possible to allocate for traveling to the office. It seems as though the action of driving to the office, insofar as it is the means to the goal and not the goal itself, is to be shrunken by the presence of other objectives. In the everyday consciousness of humans however, we do not hold every single task which we wish to complete in the mind as a conscious consideration in order to perfectly calibrate the time which we are allowed to allocate to each individual task which we need to perform throughout the day. Considering the very large number of things which people do in a single day, any attempt at this kind of analysis would be beyond our capacities and crazy. 
Instead of attempting to accomplish this, which would be very complicated and hard (and never finally converging on a definite single solution to the question of which course of action we should take to bring about our goals), it might be said in initial approximation that we simply adhere to the principle that every task in which we are involved should be completed in the shortest amount of time possible. But more than that, we seek those methods of doing things which allow us to spend more time on the things which we find happiness in, that is, we try to maximize the time we spend on those things which make us happy. In this case, the PLA seems to be expressible thus: Given any set of activities which are required to obtain an objective, the time spent on the required activities are minimized and the time spent on the objectives (the things which make us happy) are maximized.
Questions on PLA in general
The PLA seems to be a principle which is conceived of as determining the character of the third reason (the reason as to why the change occurs from one state to another). We seem to be obliged to emphasize the “character of the reason” because, as we have seen above, the PLA seems not to be a rule about how different measurable quantities change with respect to each other, although it is expressed that way sometimes, but rather, a general principle whose expressions in different kinds of changes will be different. Do we admit then, that the PLA does not cause changes to occur, but only constricts changes, so to speak, in a way which is implied by that which does cause the change to occur?
PLA within changes which are not quantitative in nature
This question posed from what has preceded above, I would like to elaborate an idea which may be seen as radical, but yet, rings with a certain note of truth: The PLA in physics is a concept which applies to those changes which are associated with quantities whose magnitude can be increased or decreased ie. energy, distance etc. However, these changes are not the only kinds of changes which exist in the universe- there are other kinds of changes even more important, which repose in realms unseen. How might we consider those kinds of changes which cannot be considered in terms of more or less?
Lets examine a process of change which we all know and love: Music. Lets try to analyze music from the standpoint of the three necessary features of change with an example. Beethoven has a final objective, it is a state in which you, the audience, understand a certain profound idea. Beethoven understands your current state of understanding. Beethoven knows how to change your understanding such that your present understanding is changed into a new understanding which he intends. This is the change of creative discovery. There is a process of variation involved in this change from beginning to end. How then does the PLA apply to this situation? What is the action which is minimized? Perhaps it is time? Lets see… no that cannot be the case, for we see that the musical piece could have been performed much quicker such that we could hear all the same exact notes and variations in a shorter interval of time, but, if we were to do this, then the musical idea would not have been conveyed! The intended change would not have occurred. In fact, it could be said that it was not even music then. Similarly, if we asked our performers to play the piece and all of its dynamic variations exactly the same but this time with a much slower tempo then, once again, we would not reach get the idea. What about volume? Well if that is minimized then we don't hear anything. If is maximized then we don't hear anything ever again. How about human effort? If we minimize the amount of effort which the players exert in the performance then the sound is dull, if it is maximized, then it is Wagnerian and ugly. So what is the correct way to play the piece? What is the most reasonable way to play the piece? Does the PLA offer us any kind of useful advice? It seems that the PLA does not offer useful application here. But, what if someone were to say that in this case we are dealing with the contours of the human mind, and thus, the minimization function must be considered in those terms and not in the terms of measurable quantities. Perhaps they would entreat us to ask: what is the performance which most easily leads the mind through the process of discovery towards the principle or idea? Which performance contains the least elements of unnecessary or accidental character which may distract the listener from the essential content? Which performance maximizes the presence of sensuous characteristics which compel the mind to stay on the path towards discovery? Is this not what characterizes one performance of music as being better or worse than another piece of music- this least action characteristic? These questions lead us to see, once more, that the PLA seems to hold in this kind of change as well.
It could be said that the case of music is somewhat different however, because in this case the division between the end result and the process by which we achieve the end result is rather blurry. The mental participation in a musical piece itself seems to be a very primary goal for the great composer, as opposed to having a single static objective such as the ones which we are familiar with in everyday life, like the static objective of obtaining eggs at the grocery store, or moving a particle from one place to another. In any case, we now have a clearer understanding of what music is. It is not sounds. For as we saw, the sounds can remain the same at different tempos and the music is lost. Music is the specific, principled, change in the mind elicited by sounds organized in a certain way. This seems to indicate again however that the PLA is an artifact of the principle of Necessary and Sufficient Reason- The least action performance is the one in which every element of the performance exists in the way that it does for the reason of conveying the precise idea.
Concept of “necessary cadence”
“Necessary cadence”, or “necessary time”. This is the term which I have found fitting to give to the clock time characteristic associated with a performed musical piece, it is not a function of a fixed rule- as some people try to define a musical piece as the relationship of frequency patterns to time. The necessary cadence is a projection determined by the motion of the mind- that is, by the physical time which is physically defined by the human mind’s process of change through musical conceptualization. Restated: The relative clock time into which such (above referenced) physical time is projected sensuously is called the “necessary cadence”. I find this term fitting for the clock time expression of the musical piece because the necessary cadence is required for the communication of the idea (as we saw above, if it is too fast or slow the discovery of the musical idea will not occur), it is required if the audience is to make the discovery, it is not arbitrary. Also, it reminds us that the music can never be defined as the necessary cadence (as the pattern of sounds in clock time) but must be defined as the substance which necessarily manifests itself in that necessary cadence in clock time in order to lead the audience to discovery. If we consider this difference between this physical time, and the necessary cadence (or expressed clock time) of the musical piece we seem to open up new horizons of investigation with respect to other physical processes. How might we apply this notion to physical systems generally? We seem, once again, to approach the issue which was raised earlier concerning the changes in the underlying “physical geometry” of different systems in the universe.
Application to physics generally
Without proceeding into too many examples let us take this analogy- In music, least action is not time, since the music is tempo sensitive (as we saw above). The change induced within the mind is primary, the capacity for the mind to recognize the principle is primary and this is what determines the necessary cadence (the tempo, or clock time). Another physical process which comes to mind as similarly tempo is Riemann's shockwave. The formation of the new physical geometry of a shockwave is also rate dependent. Too slow of a motion will not produce this kind of change. A certain relative rate in clock time, a certain necessary cadence so to speak, is required for this change to take place. What if we were to say that, just like the human mind, the universe itself goes through changes of this kind ( “makes discoveries”) under specific physical conditions, particularly with respect to time (like the necessary cadence of music)? If we were to admit this to be true, then it seems that we would need to conceive of the PLA in the form which it takes in artistic communication as the constraining principle of physical processes generally. This raises deep questions of ontology. For if we admit that our own creative mental changes are the changes to which the PLA relates in its constraining influence on physical processes other than our mental changes, then do we also admit that the changes in the universe outside our experience are also caused by agencies which are similar to our own cognitive agencies of change? What capacities, what ontological characteristics, should we attribute to the universe such that they are similar to the causes which govern human artistic action (and general action)?
The PLA is therefore seems, again, reliant on the principle of necessary and sufficient reason. See: From an initial state to a final state, there is a transformation at every smallest interval, the transformation must coincide with the final objective. Every smallest interval of action must exist for the reason of eliciting the effect desired. All features of the process which do not correspond to this requirement are to be understood as being caused by something else, they are accidental features of the process. Thus, a scantily clad theater set hosting a Shakespeare play is much more reasonable than a hollywood movie which has all the special effects etc. which distract you from real discovery, and which are in fact produced so that your mind need not to imagine anything. Here it seems that we return once again to the notion of the ontological transfinite: Must we say that the effect elicited in the mind which properly perceives a “necessary cadence” shares an identity with the very cause of that necessary cadence in the universe itself- whether in human actions or in physical processes generally? Are the causes of changes in the universe in seemingly dead and mathematical processes actually of the same character as the causes which bring into existence great works of art, or bring into existence qualitative changes in the economy? It seems that we have no choice but to consider this seriously.
“Necessary Cadence, and the Ultimate Understanding
When we seek out a necessary cadence we seek out those features of the process which most clearly express this fundamental characteristic, or principle. The other features of the process are considered to be less important,(even though they are also determined by the fundamental characteristic) by virtue of the fact that they do not generally lead us to a recognition of the process as a whole when we consider them. This calls to mind the issue which Leibniz and Cusa addressed concerning the way in which everything in the universe affects everything else. We could imagine that if our understanding were infinite as is God’s then we could understand the entirety of the universe, past present and future, by looking at any one single part of it. However, if we look at one single part of the universe with infinite Understanding, then it seems to necessarily follow that we would not see anything at all- that is, no “thing”, separate from anything else, would be seen, (since perception is based on judgements which are based on understanding). We find this to be still necessary, even through a different line of reasoning which has been shown to us by the likes of Cusa, based on the conclusion that in the Maximum understanding there is no division or separation. Therefore, someone with Maximum Understanding would not perceive (since perception is a function of understanding) anything at all, since any”thing” implies a separation of concepts- that which is the thing, that which is not the thing, and the relation between the two. It therefore seems necessary that the existence of the universe known to man must always be intermixed with a degree of confusion, as Cusa seems to also say. I would like to deepen my understanding of this. I find it interesting that this conclusion seems to follow from the investigation of the PLA we have had here.
 In fact, I do not think that it is possible to even conceive of a physical system of finite physical parameters without admitting the least action configuration implied by those parameters. We seem to be obliged to admit that all conceivable physical systems are governed by this (granted we do not admit the idea that nothing comes from nothing, such as perpetual motion machines, infinite energy etc.).
 In fact, Lyn used this kind of approach, as far as I can tell, in his elaboration of the proper construction of new cities in So You Wish.
 Take for example the seemingly intractable problems of international conflict which previously seemed to required an “infinite amount” of energy, diplomatic and otherwise, in order to resolve them- look at how these problems are now dissolving, like salt into water (to the shock of many so called historians and “experts”), as a result of the radiating influence of the BRICS process. Deeply rooted and embittered conflicts such as Russia/China, India/China, but also, most strikingly, the recent dramatic diffusions of the conflicts of China/Japan and North Korea/South Korea. How much effort, or “force” would have been required to effect such changes within the domain of the previous cultural “geometry”?
 Stated in other words: How do we induce the universe to make a discovery?What are the methods by which this can be accomplished? What is it that we are even trying to accomplish? We have been told that it is the higher principles which determine the actions which we can observe and measure. Also, that a characteristic of the universe akin to an intention as conceived of in human mental processes is in fact the same kind of principle governing the behavior of physical systems in general. Well, if the intention, or principle, is governing the behavior of the system under consideration how are we to bring the system into the new space of a new intention, a new principle, which changes the physical relationships which are possible? Moreover, how will the system, or perhaps the principle itself, know when to "make the discovery" so to speak and evolve into a higher system of physical space? We see how this can work in the case of human psychology in the most fundamental sense of human psychology, but we can conceive of how this takes place in a simple way due to the common experience, which all humans have had, of a creative discovery coming out of the arrangement of sensible entities in a particular way- visual or auditory harmonic proportions for example. But what would we say about what are generally termed "inanimate processes"? How do inanimate systems make discoveries? How do they know, for example, that the arrangements of the elements of a physical system are in such a harmonic alignment so as to allow for a new tendency to be unleashed? This is not a new question in its general form. Kepler seemed to pondered similar questions in his New Astronomy in which he asked the question as to how the planets might sense their own distance from the sun, perhaps by detecting their observed diameter of the sun.
 I am not implying that progressive leaps in fuel sources are not useful and necessary. In fact, I believe that we would be safe in assuming that the capability to induce such changes in physical geometry on larger and larger scales would require greater and greater energetic capabilities.
 I have touched upon trace aspects of this in a paper published titled “Viruses Show Us the Future”.
 To the extent that the mathematics are not physically determined.
 A long discussion might be had about what exactly the future is. How do we define it? Looks hard. What if we define the future as that which has not yet come into being. Well, then we must ask what being is. Is being our experience, or is being something else? If we say that the future is that which we will experience, then my shoes are part of the future, since, of course, I do not experience my shoes right now (I don't see them anywhere). My refrigerator is also in the future, because I do not see it now in my bedroom. Should we really say that these entities exist in the future? We feel compelled to say that they exist in the present even though we do not experience them. OK, so the present includes some things which we cannot experience yet, but so does the future. Is there any difference between something not experienced which is defined to exist in the future by the mathematical laws of physics and something else which is defined as existing in the present by those same laws but which is also not immediately experienced? There seems to be no difference save one- the ability to walk to those objects defined as presently existing and experience them when we hadn’t experienced them before. You don't really seem to be able to walk around in time like you can in space, although you can accelerate your motion into the future. Despite this however, we could still say that time was simply another existing dimension which contains things which exist just as much as the things in the perceived present exist in our three dimensions of “space”. This is actually the conceptual framework which many physicists apply in their work- a 4 dimentional space. And why should this be surprising?- This is directly implied by the nature of the concept of mathematical laws of action. LaPlace understood this consequence of his tormented theory :”...for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”
 To the extent that they are arbitrary- not physically determined.
 From this it may be asked: “Where then does this future conception lie if not within the current system of existence?” It seems that we must say that this future exists within a domain which forever transcends the totality of all existence- a place called the human imagination.
 A thorough differentiation of the general concept of potentiality seems necessary.
 One hang up which I have on this is the following: The PLA in modern physics still seems to rely on the concept of forces because, even though it has been said that the PLA eliminates the need to formulate laws in terms of force, it still utilizes the term “energy”, which, by its physics definition, is reliant on the notion of force.
 Once again, I use the term mathematical laws of physics to denote those laws which LaPlace’s Demon would make use of- those laws which define the future.
 Admittedly, the boundary between the second and first ways in which the PLA can be conceived of are somewhat blurry, but it seems useful for the investigation to separate them, at least temporarily.
 Thus, given our observations of bending light, in order for our principle to hold, physical space-time must be conceived of as being non-homogenous
 According to my calculations, it would require a human being about 1 million joules of food energy in order to walk from Boston to Seattle. By comparison, it requires about 6 Billion joules to fly a passenger aircraft a similar length. Also, multiplying these values by the relative time shows that energy-seconds is also not less by taking a plane. I didn't look at the “action” integral though.
 As implied above: I believe the attempts to characterize the motion of light as akin to the trajectories of objects in space, that is, without resorting to such rules of motion as have been discussed for light, and relying solely on mechanical notions, is not useful for many reasons, the inability to account for the least time characteristics of light using mechanical concepts being one.
 Not to mention that we also would never come to a necessary reason to take one particular path as opposed to the other if both paths required travel times which fell within the allowed limits for the travel determined in this way. Thus, even if we did this, our actions could never correspond to the principle of necessary and sufficient reason.
 Just for clarification- this is not the same as scheduling. This is the act of trying to find a reason as to which path should be taken in any given objective on the basis of attempting to determine how much time is allowed for the task. Scheduling, on the other hand, assumes beforehand an allocated time which we have already determined as being the minimum required to perform the task for specific reasons etc.
 It is true, that sometimes a new idea can be elicited as a result of changing the tempo, but this is why it is important to consider the intention of the composer.
 I am sure many more, and perhaps better examples can be found.
 Indeed, this example is not a direct correlation since any speed above the speed of sound will produce this change- that is, you can't go too fast as you can with a musical piece, but hopefully the analogy is still useful is illustrating how this kind of approach may be taken to physical systems generally, as opposed to human artistic creation alone.