In teaching, or as a student in physics, oftentimes a difficulty becomes a motivation for new understanding. In this context, what difficulty do you see in using Lagrangian or Hamiltonian methods in physics, also thinking of avoiding difficulties ahead, for example, in teaching or learning Quantum Mechanics?
As a reference, please read the following. "Consider the system of a mass on the end of a spring. We can analyze this, of course, by using F=ma to write down mx'' = −kx. The solutions to this equation are sinusoidal functions, as we well know. We can, however, figure things out by using another method, which doesn’t explicitly use F=ma. In many (in fact, probably most) physical situations, this new [150 years old] method is far superior to using F=ma. You will soon discover this for yourself when you tackle the problems and exercises for this chapter [see instructions below, or search in Google]. We will present our new [150 years old] method by rst stating its rules (without any justi cation) and showing that they somehow end up magically giving the correct answer. We will then give the method proper justification.", in Chapter 6, The Lagrangian Method, Copyright 2007 by David Morin, Harvard University.
Morin continues, "At this point it seems to be personal preference, and all academic, whether you use the Lagrangian method or the F = ma method. The two methods produce the same equations.However, in problems involving more than one variable, it usually turns out to be much easier to write down T and V , as opposed to writing down all the forces. This is because T and V are nice and simple scalars. The forces, on the other hand, are vectors, and it is easy to get confused if they point in various directions. The Lagrangian method has the advantage that once you’ve written down L ≡ T − V , you don’t have to think anymore."
instructions: search in Google, or please write requesting the link.
Something I entirely agree with. Attempting to do the double pendulum in both ways is usually a convincing argument for using the Lagrange formalism.
Dear Ed,
the Euler-Lagrange / Hamilton then Jacobi formalism is very sophisticated, elegant and powerful, based on the stationary action.
In order to fully understand the Physics, the old approach is necessary (F=ma and vectors) , otherwise the above would look like "just do the calculations" , the conservation laws which are embodied in the principle, have to be fully understood prior taking that approach.
Choose some complicated configuration and show that the only reasonable way to solve it is with that formalism.
Hello all,
Thank you for the answers.
If we call the set of problems that can be solved using vectors and conservation laws by C, and the set of problems that can be solved using vectors and Newton laws by N, we see that C > N by an unbounded amount. All of atomic physics, for example, is out. Therefore, while it seems that we could use N to prove C, as some say, using Newton's laws first, there is no such possibility.
Using C to prove N is likewise impossible, and not because N is only valid in an inertial reference frame. If we consider N extended to non-inertial frames by means of fictitious forces, we still cannot prove:
- Newton's third law, which comes from a metaphysical assumption, and contradicts C for example, that it would be remarkable to have a movement without friction -- when no friction is the norm -- and gives the wrong intuition in QM, for example.
- Newton's second law is actually missing Hoooke's law F=-kx, fluid dynamics and EM, with F depending on velocity, and friction, all coming as external forces to be discovered, while affirming that inertial mass must be equal to gravitational mass, to make gravity an external force, and comes from a metaphysical assumption as well.
- Newton's first law is impossible to prove using C, one has to use metaphysics, and we know there is no such thing as a straight line, there is no conservation of mass, and there are curved geodesic lines, while the first law is actually due to Galileo and he was not credited by Newton.
Therefore, by analysis, we cannot deduct C from N, and any attempt to use N would bring in metaphysics as a justification, and not mention Galileo's discovery.
A solution would be to teach vectors and conservation laws, mention Newton as an expression of metaphysical belief influencing physics but to be eliminated once found, as bias, and later on introduce Lagrangian and other methods that do not use vectors. Conservation laws would remain center stage, only the formalism evolves -- not the foundation.
There is no opposing force and other subjects, not only physics, could receive the different model. One need not wait for a mythical opposing reaction, equal and instantaneous, after an action. It might be adding, not equal, and not instantaneous, as in Judo and semiconductors, for example, just to cite widely different cases, SR, GR, and QM, our current theories to understand nature from a scientific, natural approach. Not that I, personally, do not like metaphysics, but I make no such hypothesis, even negating it, in physics -- to paraphrase Newton, where he made a contribution valid until today.
Cheers, Ed Gerck.
Hello Christian and all,
First, Newton's laws depend on three different metaphysical arguments, that have all been contradictory to experiments, and a hidden physical experiment by Galileo. So, it is already an advance to depend on no metaphysical argument at all but, instead, on one simple hypothesis (action is stationary), hereafter PLA [1], to explain everything, and not to hide experiments.
And all experiments, so far, have confirmed that single hypothesis, the PLA, in any scale. We have, similarly, depended on the hypothesis that the speed of light is an invariant in any reference frame, for more than a century; we have yet no theoretical proof of it, just that it is not metaphysical, not hidden, and all experiments agree. Nature is the arbiter here.
Thus, we have the trust in our present understanding of nature to use PLA and the constancy of the speed of light, both explicitly. And also as a motivation, to find an antecedent and/or a contradiction! In physics, true means "not yet false".
In finding physical laws, Max Planck, around 1900, explained that physicists should not only look to measurements, which must be all relative, but to the absolute that hides behind them, the absolute facts that physics would dream of finding, as he exemplified with the quantum.
Noether did the same when using topology to find a reason for conservation laws, and we can use them interchangeably now -- so that a conservation law can be defined by the topology, or denied. We can find conservation laws mathematically, not just stumble upon them experimentally!
Topology even controls quantum mechanics, as shown experimentally and theoretically, e.g., in many years of dedication by Thouless et. al., with their Nobel Prize awarded in 2016, for theoretical discoveries of topological phase transitions and topological phases of matter.
I am working on where a theory using PLA may go, in denying it or providing an antecedent. I think that topology has, perhaps, a meta-role here, of setting the stage for physics. So, topology may be behind, itself, the PLA. One consequence is that the PLA may be used as a basis for revisiting Noether's theorem -- instead of using a Hamiltonian or a Lagrangian, as done today. Also, some problems cannot be described with a Hamiltonian, but need a Lagrangian. Some other problems may not be describable with either, so having the PLA directly behind Noether's theorem may be useful, and maybe, solve all current cases, and allow new cases. It will be useful to me if you heard of such approach, and what your thoughts might be on this line of research, in addition to the teaching points we have been discussing here.
Cheers, Ed Gerck
[1] PLA means Principle of Least Action, where "Least", as usual in the use of the PLA in physics, describes that the first-order change in the value of the function is zero, not that the function itself is least. Any stationary case is, thereby, included. The PLA is not, as popularly said in Wikipedia, "more accurately, the principle of stationary action", it is the same. I hope this will not be confusion point for some here.
Hello Christian and all,
Your paper was a concise addition. Your arguments seem to be motivated by the question of what we consider to be a physical principle. Interestingly, the paper begins by considering Newton's laws to be based on logic, not metaphysical principles, when E. T. Jaynes is cited, saying,
"Newtonian celestial mechanics, Relativity, and Mendelian genetics are physical theories, because their mathematics was developed by reasoning out the consequences of clearly stated physical principles from which constraint the possibilities."
And Jaynes goes on to criticize QM and QED, our most successful theory of nature.
This is a rocky start, a straw-man argument, that drives the reader to a weak argumentation line. But, you save the paper by writing, "raises the question of what we consider to be a physical principle." If we forget the outdated Jaynes, and consider your questions above, we may have a better reason to "raise the question." Please give me the academic license to do that.
As we agree, all measurements are relative. But, does it follow that we are condemned to always find relative data, as Eddington presumed? Or, as you say, to "presume a reference and that must be a conserved quantity"?
For example, you cite above the unit MKS of mass, that the "prototype "kilogram" remain (approximately) constant in time". That's exactly what is not constant (vexingly for the MKS standard folk) and not conserved, so I must have missed your point there, you may want to explain better.
But, in your paper you write, "Thus, if, as scientists, we postulate that things have reason, then this is not a physical principle but a precondition, a first principle." This phrase seems to summarize well, to me, what you say above,
"... accepting the mere fact that the world is physical, immediately (without any mathematical theorem) say that there must be conserved quantities: Without conserved quantities there would be no possible reference object for a "measurement"."
So, if we take this line of reasoning, we may be using reason to justify reason, which is totally reasonable ... but not necessarily correct, as Gödel showed (any system more complex than simple arithmetic is either incomplete or inconsistent). So, we cannot use reason as the first principle, the first filter; the arbiter of any physical theory is not human reason, but nature. It does not matter who said it, how it sounds, if it disagrees with nature, it is wrong, as Feynman said.
So, if you say that measurements presume a reference and that must be a conserved quantity, that does not work in mass, and in nature. I can measure all the masses I want, and they are not relative to any conserved quantity of mass -- it is known that mass is not conserved, and it is actually an illusion created by movement, mostly, 98%, and the rest is due to the Higgs boson.
This means that there is no absolute reference regarding mass, try as the MKS folk may. Time is also without absolute reference, not even the big-bang epoch; time can start earlier. Space also has no absolute reference, my finger could be the center of the Solar system, Galileo nowithstanding, and all equations would work fine.
And reason is not the absolute meterstick either, because Gödel already proved that any system more complex than simple arithmetic is either incomplete or inconsistent. Our "mighty" reason has limits, it is incomplete, and has no absolute reference, it is inconsistent.
An absolute road, for a first principle, is the entire sytem itself, the universe, with reasons we know we ignore, reasons we ignore we ignore, and the unknown -- we do not know if it is there, or we do not see it, or it is not there at all.
That absolute road was indicated by Max Planck, around 1900, as I cited and you copied. It was followed by Noether, and may lead us to understand more about the PLA, the principle of least action. Nature is the first principle, itself, not reason, to physics. Nature has consistently broken reason, as life breaks any novel, any work of man and woman.
Cheers, Ed Gerck
Hello Christian and all,
Yet Gödel lives on, notwithstanding Hilbert and his propositions. Logic has limits, always evolves, but is smaller than the universe. The basis of physics is, funny enough, and always has been, a gift from the Greek language -- physis. It is not logic or reason, which work, however, in our present understanding of mathematics.
Most physicists are willing to accept the existence of things that cannot be proven -- that are beyond present reason. Most mathematicians are willing to accept the existence of things without any relation to phenomena — e.g.things that we, at present, cannot observe or construct in the physical world. Whether or not we can observe something directly, contemplating its possible existence may allow us to understand how it might play a role in how the world works. Thank you for your contribution.
Cheers, Ed Gerck
Hello Christian and all,
The existence of things that cannot be proven -- that are beyond present reason, is a day to day thought for most physicists. Einstein famously said, that "a single experiment can prove me wrong" -- not a reasoning, but an experiment.
Kekulé had a dream that solved the elusive structure of benzene, of a snake biting its own tail. Many examples, where what cannot be proven become the next step, things that seem to go beyond one's present reason. So, one cannot use reason as the first filter, but as the last seems very fine. Maybe we can agree there.
Cheers, Ed Gerck
I think the answer to the original question is quite obvious. Hamiltonian mechanics requires calculus always, Newtonian mechanics usually does not. It's that simple.
Is it easier to explain, and understand intuitively, and verify with experiments, that F = ma, or do we truly believe it is simpler to say that dp/dt = - the partial drivative of H with respect to q (where p is momentum)?
That's the answer to the original question. Now, whether, in describing very complex systems, Hamiltonian mechanics ultimately does a better job, that's another discussion.
Hello Albert and all,
I think your answer is targeting a high school audience, while we are thinking more of college. In college, all students are supposed to know calculus before Physics I, or learn it concurrently, so calculus is not a factor, but continuity in physics/engineering is, to a higher level.
However, if we also think about high school, we can add conservation laws and not talk about Newton's laws nor use calculus.
This could be a better preparation for college, and I saw it used well. The main problem I see in using Newton's laws is not F=ma, which is wrong, for example for a rocket with varying mass, but more the third law, which is based on metaphysics and gives the wrong intuition of action-reaction. It is more powerful to use conservation laws, at the very start, there is no unlearning to do later, and it's easier.
I think students can start college at around 13 years old that way, much earlier than currently 18, saving time by not studying what no one uses any more, or what is wrong. Somehow, we must tackle the need for students to know more things earlier, and I welcome your answer, to make it easier.
Cheers, Ed Gerck
Hello Christian,
Yes. Einstein had to write four doctor thesis, all rejected, until he added a sentence to the last thesis, and was accepted. That was not very rational for the others, but was for him. What is rational for one, follows reason, is not for another. This is often frustrating, but we all learn that is better -- in diversity of thoughts in STEM we benefit by having different ways to do things, even if one path would suffice. We can check and recheck in different ways, we can extend results.
Not having reason as the first filter is a logical way also to account for diversity, for intersubjectively accepting another opinion and test it by experiment. The attitude is of being open in what you accept as input, but selective in what you provide as output.
There is a mathematical reason for this, in Shannon's Tenth Theorem, which says that the addition of a correction channel with enough capacity can correct the errors to a value as small as we please. This surprising result, of reaching zero "as we please", is the basis of reading a disk without errors, and today's communication. Mutatis mutandis, in natural science, the correction channel is nature. It has more capacity than any other, including human reason.
Of course, that is a physicist's view. To a mathematician, nature is not important, his reason is. That diversity of reason, though, helps each other. The dichotomy was humorously explained by Feynman in https://youtu.be/MZZPF9rXzes
Cheers, Ed Gerck
Hello Christian and all,
As I see, to a physicist, belief is the probability that the evidence supports the claim (Dempster–Shafer). Reason, though, being deterministic and incomplete if consistent (Gödel), cannot stand on the way as the first filter. We would only calculate what is already knowable. It can, and may, stand in the way as the last filter. The role of intuition is to see what we don't know, cannot reason about, not only as a subconscious process in the mind.
All measurements in physics are relative, including the speed of light, but the calculations give the speed of light a value that is the same in all reference frames, hence, it is called absolute. That used to be beyond reason, some 100 years ago, with Newton's concept of time.
That value is c=1, in a proper unit system, without error or uncertainty, it is an integer number, it in adimensional; there is no error in c, or lack of constancy.
My comments on the lack of constancy of the MKS units of mass, length, and time, are not about any personal belief or quote of yours.
They originate from the currently-known technical impossibility of the French proposal of S.I., that the world accepted, heartily, today except the USA that continues with the British system. With different units for E and B, we end up measuring the same EM field with pre-Maxwell units. That and other problems, of constancy and reproducibility in MKS, are well-known and of interest in physics. The public is oblivious to these discussions.
But they illuminate the reason, ultimately, why it will fail, it has to do with this thread, and eventually a change will come. Several proposals are in the works, we already have many CGS systems. Like we say in the Internet, there are many standards to choose from. Physics can follow the proper system proposed around 1900 by Max Planck, and its variants, that does not depend on humans trying to define mass, length, and time.
I hope this helps clarify what, for brevity, I may not have said explicitly.
Cheers, Ed Gerck
Dear all,
I take advantage of the following sentence extrapolated from a thread of Christian.
"I entirely agree with him: The reference quantities have to "emerge" from theory. And my thesis is that they can only emerge as constants of motion. (The clock indeed, can not emerge as a constant of motion...)"
On the opportunity to adopt the Least Action, very powerful but, in my point of view, which did not receive the due attention since it has not been sufficiently understood (being regarded as some Methaphysical), I would like to point out on simple fact about the clocks measuring of time (which should be actually strictly connected with the least action).
At the moment it has never given a Physically acceptable/satisfactory explanation on why twin (identically constructed) atomic clocks present a time dilation though it is a fact that they present it.
SR has two ways to put clocks in desync: relativity of simultaneity (internal sync) with no proof, and lorentz factor (external sync) with a preferred frame (proven experimentally). The second can be related to a relativistic Lagrangian energy and eventually to the action principle ... GR has one way and it is beautifully modeled by the Schw solution which can be related to a generalized relativistic lagrangian and eventually the action(only Feynman did it so far). No Physical explanation thought since clocks for relativists run smoothly always at the same rate....
Dear Christian,
Most physicists accept what they cannot prove, and mathematicians are comfortable in accepting what does not exist; there are several jokes about it. There is also "expressive behavior", when rational people do what is irrational, partly explained by Freud, and exploited in marketing with "hidden persuaders". I welcome your observation, and it is a fact of nature -- that what seems to be incompatible, becomes convergent, for example in Cesaro summation, the first rigorous method for summing divergent series.
Cheers, Ed Gerck
Hello all,
One of the main difficulties in students using Hamiltonian and Lagrangian methods, is not mathematics at all, but conceptual. In fact, knowing much mathematics is detrimental, we observe, maybe caused mainly by trying to use the same tools everywhere, where different tools are needed, learning something new.
We have not yet approached much this side. Writing that electromagnetism does not accurately describe our world, in an influential paper of 1933, Bohr and Rosenfeld argued that there can be no consistent theory of the interaction between charged quantum particles and a classical electromagnetic field. This provides a good metaphor. We are investigating here that Newton's laws does not accurately describe our world. What are the consequences?
I reproduce below the arguments by Gordon Belot, in Brit. J. Phil. Sci. 49 (1998), 531-555, on electromagnetism, mutatis mutandis.
Our world could not possibly contain the sort of equations of motion described by Newton’s laws -- Newtonian mechanics is a false theory.
Now, there is a very straightforward sense in which a false -- but eminently useful theory like Newtonian mechanics could tell us about our world; it makes empirical predictions which are very accurate within a certain circumscribed domain of applicability. This has been pointed out here, and we remarked that they may seem accurate but does not explain even the field B of a permanent magnet, resting on your hand, or flying at v=0.7c.
Further, it seems strange to say that the interpretation of such a theory tells us about our world. To interpret a theory is to describe the possible worlds about which it is the literal truth. Thus, an interpretation of Newtonian mechanics seems to tell us about a possible world which we know to be distinct from our own.
On the other hand, whatever world Newtonian mechanics is true of, it is not one which contains action without reaction, as quantum mechanics affirms. So, it is difficult to see how a Newtonian mechanics effect can teach us anything about the interpretation of quantum mechanics. Of course, quantum mechanics itself is false (being nonrelativistic). So our world is one about which neither Newtonian mechanics nor quantum mechanics is true.
Nonetheless, some maintain, we learn something about our own world when we study the interpretative interaction between these two false theories.
On the other hand, not everyone wants to "follow the journey" and take something true and something false, to separate later. Some prefer to take a shortcut, and that shortcut, even to relativistic quantum mechanics, is the PLA, Lagrangian and Hamiltonian methods from the very start, after the conservation laws are learned by themselves, valid where Newtonian mechanics is not.
This conceptual view is missing to the "uninitiated", to students just starting physics. This might be the biggest obstacle -- "why learn it, if I don't need it." After all, Wikipedia says, "No new views on physics are necessarily introduced in applying Lagrangian mechanics compared to Newtonian mechanics.", and this false view cannot be edited by people (note that "necessarily" was added, after a long editing fight, but the first "No" should be edited out), one cannot change it back --please try, if you don't believe it. It is a consistent error in Wikipedia, one of many I reported before, i.a. caused by many claques of editors in futile "editing wars", that beset Wikipedia [1], to define what is reality.
Cheers, Ed Gerck
[1] https://www.researchgate.net/publication/286676558_The_Wikipedia_Experiment_falsification_and_knowledge_decay_dynamics_in_physics_and_mathematics_online
Dear Christian,
Many questions in your post, I will try to answer concisely.
1. What could be regarded as true?
The requirement that our false theories mesh in an appropriate way (as Belot does, op.cit., as a philosopher, subjectively seen) would indeed place strong constraints upon our interpretative practice of objective nature. However, in following what I called a shortcut, one does not have to take something true and something false, to separate later. For example, one does not have to sacrifice deterministic aspects we need to preserve. For students, in a tabula rasa view, after the conservation laws in mechanics are learned by themselves, valid where Newtonian mechanics is not, what role does the theory of Newtonian mechanics has to play? There is no room for that false theory, so there is nothing to unlearn or mesh. A true theory, in the shortcut approach we defend here, is thus an emerging property, as we weed out the false theories, not try to mesh them into the fabric. In that, as said before, it is better to be discriminatory enough to reject (say)10,000 truths than accept one falsity, because we can always accept a truth later, but a single falsity will poison our logic for a long time.
2. What is not yet false?
This is a synonym for truth, in physics, NOT YET FALSE. It does not mean it will be false, it's a technical use of "yet" -- it has not happened, it is truth today.
3. This might easily lead to an endless relativism. Not sure how helpful this is to students, if you can't offer some alternative?
See previous two answers, truth exists "out there" and emerges, and is refined, not meshed in with falsity. It is similar to mining the gold of truth, we need to take away the contaminants. We should not mesh the contaminants in, as Belot [op.cit.] suggests in a latter part of the same article, even tough he correctly identified most of the issues.
4.If you [Ed Gerck] say that "most physicists accept what they cannot prove", then this is true to some degree, but one should add, that the "accepted" must be useful and/or at least plausible.
In physics, it is understood that it must be on-topic, this is actually the only requiment. It does not have to be useful (as in business), nor in the least plausible. In fact, the most interesting results in physics, were either not useful or nor plausible, sometimes neither, at the time. Good research is like information , a surpise. Predictable research is not good even for bibliographies.
5. Acceptance is (or should be) preliminary, which is proven by our discussion here.
Yes, truth, in the natural sciences and not just in physics, means "not yet false". As Einstein said, "A single experiment can prove me wrong".
To close, most physicists would agree with Belot [op.cit.], that this should start from a recognition that we have a multitude of theories on the books: classical mechanics, statistical mechanics, electromagnetism, quantum mechanics, quantum statistical mechanics, quantum field theory, special relativity, general relativity, and so on. These theories tell us about very different worlds. Some are populated by particles, some by fields. In some the spatiotemporal structure is an unchanging backdrop, in some it is an active and changing participant. However, as a philosopher, Belot is unable to decide and thinks that "it is an important fact" that we find "peaceful coexistence" rather than competition.
This, one may choose to do philosophically. however, it does not work as a rule in physics, where nature is the arbiter. Especially if you take the shortcut approach oulined above. Physics is competitive, is more similar to Judo than Aikido, looking for a metaphor, in which competitivity is what hones in the techniques, and separates what works from what does not.
Cheers, Ed Gerck
Hello,
From another thread, comes an answer that shows Newton's to not even be valid for classical siuations: https://www.researchgate.net/post/Is_Newtons_Third_Law_a_misconception_A_metaphysical_dogma#view=5ab66dc5dc332d8dff017060
Someone might say that Newton's laws do not apply in non-inertial frames, despite giving names to those false centrifugal and centripetal "forces". So, Newton's laws do not apply even in our macroscopic world, they apply in a world of their own, different from ours. Our world does not have material points or inertial reference frames.
Therefore, Newton's laws cannot be consistent with our world, and surprises will (not may) happen, even within its own supposed validity domain. For example, not only the case above, but if you fall in quicksand, or believe in "action at a distance" because that is the Newton model for action-reaction pairs, although they do occur in different objects, hence not instantaneous and so on.
it's not just Newton's laws or gravitation that does mesh with our world. It's a system that gives the wrong intuition how things work. The politest thing for a physicist, I suggest, is to walk on, and not try to make current scientific sense of it. A historian or a philosopher might see it differently, because circular arguments can at least be correct, but there is no physical sense that can come out of the non-sense, such as preset in "always action-reaction".
Conservation laws are valid in this entire universe, as we know them, so its no wonder that they are valid also in the world of Newton's laws, othwerwise we would had rejected Newton's laws at the start. In other words, it is not that we live in a world that obeys Newton's laws, is that Newton's laws may obey conservation laws, and may not, and some only pay attention when they do, exclusively.
Cheers, Ed Gerck
Hello all,
See also in continuation of this thread:
1. https://www.researchgate.net/post/Should_college_students_learn_about_electromagnetism_before_mechanics
2. https://www.researchgate.net/post/Why_is_special_relativity_still_denied_today
3. https://www.researchgate.net/post/Do_we_need_to_abandon_the_Standard_Model_in_Physics
4. https://www.researchgate.net/post/Would_it_be_more_insightful_for_students_to_learn_conservation_of_momentum_laws_before_Newtons_laws
and others.
CONCLUSION: Yes, in a physics context, what students find difficult in using Lagrangian and Hamiltonian methods is built by several factors, but the major factor seems to be that alternative non-covariant descriptions such as Newtonian mechanics seem simpler.
However, vectors are a major hassle and consume too much time, considering their minor importance today. Rather than vectors (forces) in Newtonian physics with F = ma, students can focus on the Euler-Lagrange equation, or Lagrangian, where one does not use vectors because the equations of motion are based in scalars (energy).
Further, in probably most physical situations, the Euler-Lagrange equation has been superior, for about 250 years, to using F = ma (even when F = ma is valid) and especially where it is not, such as inside an atom or near a large mass.
More at Preprint New Physics with the Euler-Lagrange Equation: Going Beyond Newton
All: The physical model behind this approach to quantum mechanics, as not tied only to the microscopic view, to an atomic view of matter that we find has further subdivisions, is the Euler-Lagrange equation, harking back to the Sturm-Liouville differential equation, but In Galois fields.
For a limited time, my home-page has two free full-text papers on it -- as offered through Amazon and Kindle books -- for any RG members, just for the asking, in PDF. Sent me please a PM, and I will send back. At least until June 30.
Recently, this question confirms that infinitesimals do not EXIST:
https://www.researchgate.net/post/Can_infinitesimals_be_eliminated_in_calculus_SOLVED2
And that, also, vectors cannot be used in physics after 2D. Scalars have been used, instead, with the Euler-Lagrange equations. This is helpful also for QM.
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