There is a fundamental principle which governs and drives all nature. This is not some pie in the sky articulation of cosmic oneness. It is in fact what actually has been shown to govern the physical laws of our universe as far as we can tell. It is called the principle of least action.

You know physics and its familiar laws:
s = vt (distance = rate * time)
F = ma (force = mass * acceleration)
KE = 1/2(mv^2) (Kinetic Energy = .5 mass times velocity squared)
p = mv (momentum = mass * velocity)

There is however an alternative formulation of physics known as Lagrangian Dynamics. Lagrangian Dynamics does not solve for answers to physics questions with the common equations directly, but rather takes the entire physical system as a whole. Lagrangian = Potential Energy + Kinetic Energy. One can determine the path particles will take in motion in reference to the total system in the aggregate.

It's almost like saying, I could calculate the trajectory of the baseball you throw using newton's laws, but instead I choose to do so by analyzing the entire stadium and looking for alterations in the entirety of the energy contained within the system.

That is possible because of one reason: Nature always chooses the path of least resistance. This is known as the principle of least action.

If I have an equation which takes into account the entire system, then within this equation, if i find the function which minimizes the trajectory, that function is the set of laws for the system.
Just like you did in algebra, finding the vertex of a parabola, or in calculus where you found the lower bound by finding the point where the derivative = 0, finding the minima of the Lagrangian will literally let you derive the entire set of the laws of physics which must govern that system.

It is an absolutely astonishing and unbridled creation of human genius or discovery. It is what Euler and Lagrange worked on, and what countless others on the way to Einstein added to. It is the backdrop of Einstein's work in discovering general relativity.

Every single law of physics that we currently know of is derivable from the Principle of least action, which mathematically translates to finding the set of functions at which the Lagrangian is a mathematical minimum.
Those of you who have taken physics know that all physics basically derives from physics. One equation leads to other equations. Even einstein's theory of relativity is mathematically derivable from this.

The question now is this: Have any of you heard of this before? Something so encompassing, so simply expressed, and yet so elegant is bound to have provoked some philosophical thought as to what it all means.
What branch of philosophy does this even fall under?

I am not at all a philosopher. But I did google to find these:

I've heard of the general many times in many ways, but Lagrangian Dynamics is a new one to me.
I'd call this metaphysics. Metaphysics deals with the fundamental natures of the world, and this seems to fit the bill.

As the question of what it means, this raises the question of whether it means anything. And to that effect, I find it difficult to say that it does mean anything. Meaning must derive from something, and in this way, not everything can be meaningful- there must be a basic foundation of things which are meaningless from which meaning derives. Such a principle as this is an easy candidate for a meaningless foundation. Unless by 'meaning' you mean 'significance' (rather than something to with purpose and cause) which is simply a relative assessment.

I've heard of the general many times in many ways, but Lagrangian Dynamics is a new one to me.
I'd call this metaphysics. Metaphysics deals with the fundamental natures of the world, and this seems to fit the bill.

As the question of what it means, this raises the question of whether it means anything. And to that effect, I find it difficult to say that it does mean anything. Meaning must derive from something, and in this way, not everything can be meaningful- there must be a basic foundation of things which are meaningless from which meaning derives. Such a principle as this is an easy candidate for a meaningless foundation. Unless by 'meaning' you mean 'significance' (rather than something to with purpose and cause) which is simply a relative assessment.

But this is more than just merely ascribing meaning to something. Why the principle of least action in the context of lagrangian dynamics is significant over a pie in the sky emotional expressions about "oneness with the universe" is that the laws of physics are actually derivable from this.

The principle of least action is essentially "nature will always choose the shortest path"
Mathematically that means taking the total derivative of the functional of the lagrangian. And from there, essentially all physical laws mathematically spring forth.

This isn't a matter of humans ascribing meaning to the principle of least action so much as it is the fundamental principle upon which the universe appears to be based.

If I tell you distance = rate * time, that isn't really up for interpretation. Distance will equal rate times time. It is a truth pervasive enough, solid and concrete enough, we can build an actual world around the implications of that. For this reason, I believe it goes beyond our human ascribing of meaning to it. Distance, a concept we defined, will equal rate * time. We may have defined all those concepts. But once we defined distance, speed, and time, the truth of distance = rate * time exists, whether we discover it or not.

Euler and Lagrange proved that Lagrangian mechanics is derivable from Newton's laws, and vice versa. If you think about this relationship carefully, it is telling you exactly and unequivocally what the metaphysical implications of the action principle are. They are the same, no more and no less, than the metaphysical implications of Newton's laws. The same goes for action principles of other theories of physics. In each case, stating the theory in terms of an action principle is exactly equivalent to stating the theory in terms of classical differential equations.

The value of the action principle is that it affords a different perspective on physical problems that is often clearer than alternatives. For instance, many problems in classical mechanics involving multiple interconnected bodies are easier to solve using Lagrangian mechanics. Converting Dirac's equation for the electron to Lagrangian form famously led Feynman to the path-integral formulation of quantum mechanics.

But in both of those cases, as well as all others without exception, the problem could have been solved using the good old "crank out the differential equations" approach -- it just would have been harder. The math is unequivocal. "Nature searches for the path of least action" and "Nature locally executes these differential equations at each instant" are logically equivalent statements.

That philosophy paper about the metaphysics of the action principle was written by a BS artist. Even if you didn't know the math, you could tell from all the weasel words and hedging. I would hold that paper up as an archetypical example of why Richard Feynman had such a contempt for philosophy.

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A limit of time is fixed for thee
Which if thou dost not use for clearing away the clouds from thy mind
It will go and thou wilt go, never to return.

>But in both of those cases, as well as all others without exception, the problem could have been solved using the good old "crank out the differential equations" approach -- it just would have been harder. The math is unequivocal. "Nature searches for the path of least action" and "Nature locally executes these differential equations at each instant" are logically equivalent statements.

But I would counter that the math being too difficult is the problem. Or rather, we cannot divine the full implications of what we know right away.
Lagrangian Mechanics is equal to Newton's laws. And sure if you think about it really hard you could arrive at that conclusion.
It just so happens that it took humankind a century to figure that out.

Why are so many differential equations unsolvable for us at the moment? Is it because we aren't trying hard enough? Or is it perhaps because the abstractions we have used to build mathematics
are themselves in someway deficient or inefficient in ways we do not yet understand?

Do you know what a quaternion is? A number of the form a + bi +cj + dk.
It's a bit of an oddball really. I'm not sure where else it's used. I use it for computer graphics rotation computations.
But if you don't use it in physics or math, there's reason for that.

From wiki. From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature of quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side-effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to understand. However, quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations.

Much of our math and physics is built on vectors and tensors. It didn't have to be that way. But ultimately the abstract concept of a vectors proved to be more useful in us describing phenomena.
That doesn't mean vectors are "correct." Quaternions regained some utility in being able to describe base phenomena in ways that were not clear using a vector description.

I should think it the same way with approaching a description of nature with newton's law vs lagrangian and hamiltonian mechanics. Perhaps someday we will derive a wonderfully more efficient abstraction upon which to base our mathematics that would result in even greater insight. But I do not think you can simply dismiss these alternative formulations as exactly and unequivocally equal if one had simply thought hard enough about it.

While true, the whole point is that alternative formulations give us insight on nature in ways that a single formulation would not. The value is there because we aren't smart enough to figure out all of nature with just an algebraic/calculus/differential equation formulation.

I end my point with a reference to Fermat's last theorem.
no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

If one simply thought hard enough about this, you are correct they would have been able to figure out a proof, or a set of logical steps that would establish that statement as true.
It took humanity 358 years to find an answer to that question. And by the time we had, we had created entirely new branches of mathematics describing properties of spaces and other mathematical abstractions
(galois theory) that made the proof possible.

it is because mentally divining the equivalence of newton's laws and a principle of least action requires so much thought (years for Euler and Lagrange) that there is value to be had--insight perhaps in the physical phenomena both hope to describe.

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There is a fundamental principle which governs and drives all nature. This is not some pie in the sky articulation of cosmic oneness. It is in fact what actually has been shown to govern the physical laws of our universe as far as we can tell. It is called the principle of least action.

You know physics and its familiar laws:

s = vt (distance = rate * time)

F = ma (force = mass * acceleration)

KE = 1/2(mv^2) (Kinetic Energy = .5 mass times velocity squared)

p = mv (momentum = mass * velocity)

There is however an alternative formulation of physics known as Lagrangian Dynamics. Lagrangian Dynamics does not solve for answers to physics questions with the common equations directly, but rather takes the entire physical system as a whole. Lagrangian = Potential Energy + Kinetic Energy. One can determine the path particles will take in motion in reference to the total system in the aggregate.

It's almost like saying, I could calculate the trajectory of the baseball you throw using newton's laws, but instead I choose to do so by analyzing the entire stadium and looking for alterations in the entirety of the energy contained within the system.

That is possible because of one reason: Nature always chooses the path of least resistance. This is known as the principle of least action.

If I have an equation which takes into account the entire system, then within this equation, if i find the function which minimizes the trajectory, that function is the set of laws for the system.

Just like you did in algebra, finding the vertex of a parabola, or in calculus where you found the lower bound by finding the point where the derivative = 0, finding the minima of the Lagrangian will literally let you derive the entire set of the laws of physics which must govern that system.

It is an absolutely astonishing and unbridled creation of human genius or discovery. It is what Euler and Lagrange worked on, and what countless others on the way to Einstein added to. It is the backdrop of Einstein's work in discovering general relativity.

Every single law of physics that we currently know of is derivable from the Principle of least action, which mathematically translates to finding the set of functions at which the Lagrangian is a mathematical minimum.

Those of you who have taken physics know that all physics basically derives from physics. One equation leads to other equations. Even einstein's theory of relativity is mathematically derivable from this.

The question now is this: Have any of you heard of this before? Something so encompassing, so simply expressed, and yet so elegant is bound to have provoked some philosophical thought as to what it all means.

What branch of philosophy does this even fall under?

I am not at all a philosopher. But I did google to find these:

https://arxiv.org/ftp/arxiv/papers/1511/1511.03429.pdf

https://egtheory.wordpress.com/2014/09/28/principle-of-least-action/

Is this metaphysics? Is this mathematical philosophy? What do you think about this if you have never heard of this before?

I'd call this metaphysics. Metaphysics deals with the fundamental natures of the world, and this seems to fit the bill.

As the question of what it means, this raises the question of whether it means anything. And to that effect, I find it difficult to say that it does mean anything. Meaning must derive from something, and in this way, not everything can be meaningful- there must be a basic foundation of things which are meaningless from which meaning derives. Such a principle as this is an easy candidate for a meaningless foundation. Unless by 'meaning' you mean 'significance' (rather than something to with purpose and cause) which is simply a relative assessment.

Custom Sets- All feedback is appreciated!RUNIN: Norse mythology set (awaiting further playtesting)

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Contibutor to the Pyrulea community set

I'm here to tell you that all your set mechanics are bad

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But this is more than just merely ascribing meaning to something. Why the principle of least action in the context of lagrangian dynamics is significant over a pie in the sky emotional expressions about "oneness with the universe" is that the laws of physics are actually derivable from this.

The principle of least action is essentially "nature will always choose the shortest path"

Mathematically that means taking the total derivative of the functional of the lagrangian. And from there, essentially all physical laws mathematically spring forth.

This isn't a matter of humans ascribing meaning to the principle of least action so much as it is the fundamental principle upon which the universe appears to be based.

If I tell you distance = rate * time, that isn't really up for interpretation. Distance will equal rate times time. It is a truth pervasive enough, solid and concrete enough, we can build an actual world around the implications of that. For this reason, I believe it goes beyond our human ascribing of meaning to it. Distance, a concept we defined, will equal rate * time. We may have defined all those concepts. But once we defined distance, speed, and time, the truth of distance = rate * time exists, whether we discover it or not.

exactly and unequivocallywhat the metaphysical implications of the action principle are. They are the same, no more and no less, than the metaphysical implications of Newton's laws. The same goes for action principles of other theories of physics. In each case, stating the theory in terms of an action principle is exactly equivalent to stating the theory in terms of classical differential equations.The value of the action principle is that it affords a different perspective on physical problems that is often clearer than alternatives. For instance, many problems in classical mechanics involving multiple interconnected bodies are easier to solve using Lagrangian mechanics. Converting Dirac's equation for the electron to Lagrangian form famously led Feynman to the path-integral formulation of quantum mechanics.

But in both of those cases, as well as all others without exception, the problem could have been solved using the good old "crank out the differential equations" approach -- it just would have been harder. The math is unequivocal. "Nature searches for the path of least action" and "Nature locally executes these differential equations at each instant" are logically equivalent statements.

That philosophy paper about the metaphysics of the action principle was written by a BS artist. Even if you didn't know the math, you could tell from all the weasel words and hedging. I would hold that paper up as an archetypical example of why Richard Feynman had such a contempt for philosophy.

A limit of time is fixed for theeWhich if thou dost not use for clearing away the clouds from thy mind

It will go and thou wilt go, never to return.

But I would counter that the math being too difficult

isthe problem. Or rather, we cannot divine the full implications of what we know right away.Lagrangian Mechanics is equal to Newton's laws. And sure if you think about it really hard you could arrive at that conclusion.

It just so happens that it took humankind a century to figure that out.

Why are so many differential equations unsolvable for us at the moment? Is it because we aren't trying hard enough? Or is it perhaps because the abstractions we have used to build mathematics

are themselves in someway deficient or inefficient in ways we do not yet understand?

Do you know what a quaternion is? A number of the form a + bi +cj + dk.

It's a bit of an oddball really. I'm not sure where else it's used. I use it for computer graphics rotation computations.

But if you don't use it in physics or math, there's reason for that.

From wiki.

From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature of quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side-effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to understand. However, quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations.Much of our math and physics is built on vectors and tensors. It didn't have to be that way. But ultimately the abstract concept of a vectors proved to be more useful in us describing phenomena.

That doesn't mean vectors are "correct." Quaternions regained some utility in being able to describe base phenomena in ways that were not clear using a vector description.

I should think it the same way with approaching a description of nature with newton's law vs lagrangian and hamiltonian mechanics. Perhaps someday we will derive a wonderfully more efficient abstraction upon which to base our mathematics that would result in even greater insight. But I do not think you can simply dismiss these alternative formulations as exactly and unequivocally equal if one had simply thought hard enough about it.

While true, the whole point is that alternative formulations give us insight on nature in ways that a single formulation would not. The value is there because we aren't smart enough to figure out all of nature with just an algebraic/calculus/differential equation formulation.

I end my point with a reference to Fermat's last theorem.

no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.

If one simply thought hard enough about this, you are correct they would have been able to figure out a proof, or a set of logical steps that would establish that statement as true.

It took humanity 358 years to find an answer to that question. And by the time we had, we had created entirely new branches of mathematics describing properties of spaces and other mathematical abstractions

(galois theory) that made the proof possible.

it is because mentally divining the equivalence of newton's laws and a principle of least action requires so much thought (years for Euler and Lagrange) that there is value to be had--insight perhaps in the physical phenomena both hope to describe.