Cleonis

About the stationary action concept: Yeah, it looks impenetrable, but here's the thing: there is a way of looking at it from just the right angle, and then becomes transparent.

Part of the story is this: the actual criterion is: the true trajectory corresponds to a point in variation space where the derivative of the action (derivative wrt applied variation) is zero.

In the cases examined when the concept was first introduced I suppose that in those cases the derivative-is-zero point was seen to be a minimum. From there, I suppose, came a supposition that there was some form of minimization at play.

However, within the scope of classical mechanics there are also classes of cases such that at the point in variation space corresponding to the true trajectory the action is at a maximum.

The above, and other aspects, are discussed in a resource that I created.

https://cleonis.nl/physics/phys256/energy_position_equation....

In the resource the mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond.

About interpretation: As we know: motion along the true trajectory has the property that at every point in time the rate of change of kinetic energy matches the rate of change of potential energy. As we know: that property is known as the work-energy theorem.

The criterion derivative-wrt-variation-is-zero corresponds mathematically to the property: rate-of-change-of-kinetic-energy-matches-the-rate-of-change-of-potential-energy.

In the resource a two stage process is presented:

- Derivation of the work-energy theorem from F=ma

- Transformation from the work-energy theorem to classical mechanics stationary action

Of course: when you look at the work-energy theorem you wouldn't expect that it can be transformed to classical mechanics stationary action. The transformation consists of multiple steps. In the resource I present it step by step; for each step the logic and consistency is readily recognizable.

For me, having the breakdown into mathematical elements available changed my whole perspective on classical mechanics stationary action.

I hope I can persuade you to check out the resource

As to understanding Hamilton's stationary action deeply: that is accessible.

I have created a resource with interactive diagrams. Move sliders to sweep out variation of a trial trajectory. The diagram shows the response.

https://cleonis.nl/physics/phys256/energy_position_equation....

About the form of the resource:

In physics textbooks the usual presentation is to posit Hamilton's stationary action, followed by demonstration that F=ma can be recovered from it.

Now: we have that in physics you can often run derivations in both directions.

Example: the connection between the Lagrangian formulation of mechanics and the Hamiltonian formulation. The interconversion is by way of Legendre transformation. Legendre transformation is it's own inverse; applying Legendre transformation twice recovers the original function.

Well, the relation between F=ma and Hamilton's stationary action is a bi-directional relation too: it is possible to go _from_ F=ma _to_ Hamilton's stationary action.

The process has two stages:

- Derivation of the work-energy theorem from F=ma

- Demonstration that in circumstances such that the work-energy theorem holds good Hamilton's stationary action holds good also.

Knowing how to go from F=ma to Hamilton's stationary action goes a long way towards lifting the sense of mystery.

General remark: Of course, in physics there are many occurrences of hierarchical relation. Classical mechanics has been superseded by Quantum mechanics, with classical mechanics as limiting case; the validity of classical mechanics must be attributed to classical mechanics emerging from quantum mechanics in the macroscopic limit.

But in the case of the relations between F=ma, the work-energy theorem, and Hamilton's stationary action: the bi-directionality informs us that the relations are not hierarchical; those concepts are on equal par.

About transitioning from Classical Mechanics to QM, guided by observations.

There is a very interesting approach in the quantum physics book by Eisberg and Resnick, section 5.2

To arrive at the Schrödinger equation Eisberg and Resnick construct what they refer to as a plausibility argument.

The goal: to arrive at a wave equation that when solved for the Hydrogen atom will have the electron orbitals as set of solutions.

Eisberg and Resnick state 4 demands:

-1. Must be consistent with the de Broglie/Einstein postulates. wavelength=h/p, frequency=E/h

-2. Must be such that for a quantum entity followed over time the sum of potential energy and kinetic energy is a conserved quantity.

-3. Must be such that the equation is linear in \Psi(x,t): any linear combination of two solutions \Psi_1 and \Psi_2 must also be a solution of the equation. (Motivation: in experiments electron diffraction effects are observed. Interference effects can occur only if wave functions can be _added_.)

-4. In the absence of a potential gradient the equation must have as a solution a propagating sinusoidal wave of constant wavelength and frequency.

Eisberg and Resnick proceed to show that the above 4 demands narrow down the possibilities such that arriving at the Schrödinger equation is made inevitable.

To me the second demand is particularly interesting. The second demand is equivalent to demanding that the work-energy theorem holds good. The recurring theme: the work-energy theorem.

I have a (html)-transcript of the Eisberg & Resnick treatment that I can make available to you.

There is a youtube video with a presentation that is based on the Eisberg & Resnick plausibility argument.

https://youtu.be/2WPA1L9uJqo

In that video the presentation of the plausibility argument is in the first 18 minutes, the rest of the video is about application of the Schrödinger equation.

The rewrite of section 2 of the article is now pushed out to the web page.

Repeating the links: Page dedicated to the case of a potential proportional to the cube of the displacement: http://cleonis.nl/physics/phys256/stationary_action.php

From F=ma to Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...

There are other demonstrations available that go from the newtonian formulation to Hamilton's stationary action. I believe the one in my resource is the most direct demonstration. (As in: a more direct path doesn't exist, I believe.)

(If you are interested, I can give links to the other demonstrations that I know about.)

About the article with mathematical treatment: http://cleonis.nl/physics/phys256/energy_position_equation.p...

One section of that will be replaced in a day or two: the last part of section 2. I completed a new diagram, that diagram will allow me to cut a lot of text. I believe the change will be a significant improvement.

About d'Alembert's principle. A modern name for it is 'd'Alembert's virtual work'.

The modern concept of 'work done' was formulated around 1850 (Eighteen-fifty). That is, we shouldn't assume that back in the days of Lagrange d'Alembert's principle was understood in the same way as it is today.

Joseph Louis Lagrange motivated his notion of potential energy in terms of d'Alembert's principle.

The recurring theme is the concept of 'work done'.

In case you hadn't noticed yet, I'm the contributor who notified you of a resource I created, with interactive diagrams.

There is this distinction: the work-energy theorem expresses physical motion, whereas d'Alembert's virtual work expresses, as the modern name indicates, virtual work.

My assessment is that using d'Alembert's virtual work is an unnecesarily elaborate approach. The same result can be arrived at in a more direct way.

While most authors posit the stationary action concept as a given, it is in fact possible to go from the newtonian formulation to the Lagrangian formulation, and from there to Hamilton's stationary action.

That is, the relations between the various formulations of classical mechanics are all bi-directional.

At the hub of it al is the work-energy theorem.

I created a resource with interactive diagrams. Move a slider to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond to the variation that is applied.

Starter page: http://cleonis.nl/physics/phys256/stationary_action.php The above page features a case that allows particularly vivid demonstration. An object is launched upwards, subject to a potential that increases with the cube of the height. The initial velocity was tweaked to achieve that after two seconds the object is back to height zero. (The two seconds implementation is for alignment with two other diagrams, in which other potentials have been implemented; linear and quadratic.)

Article with mathematical treatment: http://cleonis.nl/physics/phys256/energy_position_equation.p...

To go from F=ma to Hamilton's stationary action is a two stage proces:

- Derivation of the work-energy theorem from F=ma

- Demonstration that in cases such that the work-energy theorem holds good Hamilton's stationary action holds good also.

General remarks: In the case of Hamilton's stationary action the criterion is: The true trajectory corresponds to a point in variation space such that the derivative of Hamilton's action is zero. The criterion derivative-is-zero is sufficient. Whether the derivative-is-zero point is at a mininum or a maximum of Hamilton's action is of no relevance; it plays no part in the reason why Hamilton's stationary action holds good.

The true trajectory has the property that the rate of change of kinetic energy matches the rate of change of potential energy. Hamilton's stationary action relates to that.

The power of an interactive diagram is that it can present information simultaneously. Move a slider and you see both the kinetic energy and the potential energy change in response. It's like looking at the same thing from multiple angles all at once.

Indeed inertia. Theory of motion consists of describing the properties of Inertia.

In terms of Newtonian mechanics the members of the equivalence class of inertial coordinate systems are related by Galilean transformation.

In terms of relativistic mechanics the members of the equivalence class of inertial coordinate systems are related by Lorentz transformation.

Newton's first law and Newton's third law can be grouped together in a single principle: the Principle of uniformity of Inertia. Inertia is uniform everywhere, in every direction.

That is why I argue that for Newtonian mechanics two principles are sufficient.

The Newtonian formulation is in terms of F=ma, the Lagrangian formulation is in terms of interconversion between potential energy and kinetic energy

The work-energy theorem expresses the transformation between F=ma and potential/kinetic energy The work-energy theorem: I give a link to an answer by me on physics.stackexchange where I derive the work-energy theorem https://physics.stackexchange.com/a/788108/17198

The work-energy theorem is the most important theorem of classical mechanics.

About the type of situation where the Energy formulation of mechanics is more suitable: When there are multiple degrees of freedom then the force and the acceleration of F=ma are vectorial. So F=ma has the property that the there are vector quantities on both sides of the equation.

When expressing in terms of energy: As we know: the value of kinetic energy is a single value; there is no directional information. In the process of squaring the velocity vector directional information is discarded, it is lost.

The reason we can afford to lose the directional information of the velocity vector: the description of the potential energy still carries the necessary directional information.

When there are, say, two degrees of freedom the function that describes the potential must be given as a function of two (generalized) coordinates.

This comprehensive function for the potential energy allows us to recover the force vector. To recover the force vector we evaluate the gradient of the potential energy function.

The function that describes the potential is not itself a vector quantity, but it does carry all of the directional information that allows us to recover the force vector.

I will argue the power of the Lagrangian formulation of mechanics is as follows: when the motion is expressed in terms of interconversion of potential energy and kinetic energy there is directional information only on one side of the equation; the side with the potential energy function.

When using F=ma with multiple degrees of freedom there is a redundancy: directional information is expressed on both sides of the equation.

Anyway, expressing mechanics taking place in terms of force/acceleration or in terms of potential/kinetic energy is closely related. The work-energy theorem expresses the transformation between the two. While the mathematical form is different the physics content is the same.

I will argue that 'has least action as foundation' does not in itself imply that Lagrangian mechanics is a sparser theory:

Here is something that Newtonian mechanics and Lagrangian mechanics have in common: it is necessary to specify whether the context is Minkowski spacetime, or Galilean spacetime.

Before the introduction of relativistic physics the assumption that space is euclidean was granted by everybody. The transition from Newtonian mechanics to relativistic mechanics was a shift from one metric of spacetime to another.

In retrospect we can recognize Newton's first law as asserting a metric: an object in inertial motion will in equal intervals of time traverse equal distances of space.

We can choose to make the assertion of a metric of spacetime a very wide assertion: such as: position vectors, velocity vectors and acceleration vectors add according to the metric of the spacetime.

Then to formulate Newtonian mechanics these two principles are sufficient: The metric of the spacetime, and Newton's second law.

Hamilton's stationary action is the counterpart of Newton's second law. Just as in the case of Newtonian mechanics: in order to express a theory of motion you have to specify a metric; Galilean metric or Minkowski metric.

To formulate Lagrangian mechanics: choosing stationary action as foundation is in itself not sufficent; you have to specify a metric.

So: Lagrangian mechanics is not sparser; it is on par with Newtonian mechanics.

More generally: transformation between Newtonian mechanics and Lagrangian mechanics is bi-directional.

Shifting between Newtonian formulation and Lagrangian formulation is similar to shifting from cartesian coordinates to polar coordinates. Depending on the nature of the problem one formulation or the other may be more efficient, but it's the same physics.

Thank you for taking the time to have a look.

About the presentation: I think I agree: once I'm up to the level of discussing Lagrangians and stationary action I should not re-teach integration; the reader will be familiar with that.

That particular presentation grew over time; I agree it is uneven. I need to scrap a lot of it.

The preceding article http://cleonis.nl/physics/phys256/calculus_variations.php Is more an overarching concept.

Also, I'm active on the stackexchange physics forum. Over the years: Hamilton's stationary action is a recurring question subject. Some weeks ago I went back to the first time a stationary action question was posted, submitting an answer. In that answer: I aimed to work the exposition down to a minimum, presenting a continuous arch. https://physics.stackexchange.com/a/821469/17198

three sections:

1. Work-Energy theorem

2. The central equation of the work 'Mécanique Analytique' by Joseph Louis Lagrange (I discuss _why_ that equation obtains.)

3. Hamilton's stationary action

It's a tricky situation. I'm not assuming the thing I present derivation of, but I can see how it may appear that way.

If I don't hear back in a week or so I will remind you, I hope that's OK with you.

I'm aware your expectations may be low. Your thinking may be: if textbook authors such as John Taylor don't know the why, then why would some random dude know?

The thing is: this is the age of search machines on the internet; it's mindblowing how searcheable information is. I've combed, I got to put pieces of information together that hadn't been put together before, and things started rolling.

I'm stoked; that's why I'm reaching out to people.

I came across the ycombinator thread following up something that Jess Riedel had written.

Hi, I want to respond to a post from you from 2019. (That 2019 thread no longer offers the reply button, otherwise I would reply there of course.) I apologize for using this thread to get my message in.

This is the item I want to respond to: https://news.ycombinator.com/item?id=19768492 When you took a Classical Mechanics course you were puzzled by the form of the Lagrangian: L = T - V

I have created a resource for the purpose of making application of calculus of variations in mechanics transparent. As part of that the form of the Lagrangian L=T-V is explained.

http://cleonis.nl/physics/phys256/calculus_variations.php

http://cleonis.nl/physics/phys256/energy_position_equation.p...

I recognize the 'you are certainly entitled to ask why' quote, it's from the book 'Classical Mechanics' by John Taylor.

Here's the thing: there is a good answer to the 'why' question. Once you know that answer things become transparent, and any wall is gone.

There is a way of _arriving_ at that subtraction, rather than just throwing it out there.

A resource I created:

Calculus of Variations as applied in physics: http://cleonis.nl/physics/phys256/calculus_variations.php

Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...

In that resource I show why it works.

In an earlier answer I gave more information about that resource. To find that earlier answer: go up to the entire thread, and search on the page for my nick: Cleonis

I have created a resource for the purpose of making Hamilton's stationary action transparent.

It is possible to go in all forward steps from F=ma to Hamilton's stationary action; that is what I present.

The path from F=ma to Hamilton's stationary action consists of two stages: (1) Derivation of the work-energy theorem from F=ma (2) Demonstration: when the conditions are such that the work-energy theorem holds good then Hamilton's stationary action will hold good also.

I recommend that you first absorb the presentation of the subset of Calculus of Variations that is applied in physics: http://cleonis.nl/physics/phys256/calculus_variations.php

Discussion of Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...

These presentations are illustrated with interactive diagrams. Each diagram has one or more sliders for manipulation of the contents of the diagram. That way a single diagram can offer a range of cases/possibilities.

About my approach: I think of Hamilton's stationary action as an engine with moving parts. To show how an engine works: construct a model out of translucent plastic, so that the student can see all the way inside, and see how all of the moving parts interconnect. My presentation is in that spirit.

In retrospect: the earliest recognition of a conserved quantity was Kepler's law of areas. Isaac Newton later showed that Kepler's law of areas is a specific instance of a property that obtains for any central force, not just the (inverse square) law of gravity.

About symmetry under change of orientation: for a given (spherically symmetric) source of gravitational interaction the amount of gravitational force is the same in any orientation.

For orbital motion the motion is in a plane, so for the case of orbital motion the relevant symmetry is cilindrical symmetry with respect to the plane of the orbit.

The very first derivation that is presented in Newton's Principia is a derivation that shows that for any central force we have: in equal intervals of time equal amounts of area are swept out.

(The swept out area is proportional to the angular momentum of the orbiting object. That is, the area law anticipated the principle of conservation of angular momentum)

A discussion of Newton's derivation, illustrated with diagrams, is available on my website: http://cleonis.nl/physics/phys256/angular_momentum.php

The thrust of the derivation is that if the force that the motion is subject to is a central force (cilindrical symmetry) then angular momentum is conserved.

So: In retrospect we see that Newton's demonstration of the area law is an instance of symmetry-and-conserved-quantity-relation being used. Symmetry of a force under change of orientation has as corresponding conserved quantity of the resulting (orbiting) motion: conservation of angular momentum.

About conservation laws:

The law of conservation of angular momentum and the law of conservation of momentum are about quantities that are associated with specific spatial characteristics, and the conserved quantity is conserved over time.

I'm actually not sure about the reason(s) for classification of conservation of energy. My own view: we have that kinetic energy is not associated with any form of keeping track of orientation; the velocity vector is squared, and that squaring operation discards directional information. More generally, Energy is not associated with any spatial characteristic. Arguably Energy conservation is categorized as associated with symmetry under time translation because of absence of association with any spatial characteristic.

I have a comment about Lagrangian models.

(I'm not commenting on the "All-at-once" angle, that is out of my league.)

You assert a contrast, with on one hand (traditional physics) tracking motion step by step, and on the other hand (Lagrangian) an approach that considers the overall path.

I will argue that in actual fact that contrast is far smaller than it appears to be.

In preparation I start with addressing the following: it is not the case that the true trajectory always coincides with a minimum of the action. There are also classes of cases such that the true trajectory coincides with a maximum of the action. Within the scope of Hamilton's stationary action there is an inversion: from classes of cases with minimum to classes of cases with maximum.

How can it be that within the single scope, here Hamilton's stationary action, both are viable?

The reason for that is: it is not about minimum nor maximum. The actual criterion is the property that the two have in common: as you sweep out variation: the point in variation space such that the derivative of the action is zero coincides with the true trajectory.

Next item in the preparation: the far reaching scope of differential equations.

When we solve a differential equation the solution that is obtained is a function. In that sense a differential equation is a higher level equation. A low level equation has a number as its solution. But a differential equation has an entire function as its solution. A differential equation states: this relation must be satisfied concurrently for all values of the domain. That is to say: when you solve a differential equation the solution that you obtain is for the entire path.

Now to the main point: Calculus of variations has a particular mathematical property, I will use the catenary problem to showcase that property. The catenary problem: what is the shape of a chain that is suspended between two points? We consider the most general case: for any height difference between the two points of suspension. We have that the resting state is a state of minimal potential energy. That is to say: for the shape of the catenary the derivative of the potential energy wrt variation of the shape is zero.

Now divide the solution in subsections. Every subsection is an instance of the catenary problem. We can solve each of the subsections, and then concatenate those subsections. We can continue the subdividing; you can still concatenate the subsolutions. There is no lower limit to the size of the subsections; the reasoning remains valid down to infinitesimally short subsections.

Given that infinitesimal property: it follows that it should be possible to solve the catenary problem with a differential equation. I have on my website a demonstration of how to set up and solve the differential equation for the catenary problem. It's in an article titled: 'Calculus of Variations as applied in physics'. http://cleonis.nl/physics/phys256/calculus_variations.php

More generally, this infinitesimal property explains why the Euler-Lagrange equation is a differential equation.

Action concepts are stated in the form of an integral, but here's the thing: the variational property obtains at the infinitesimal level, and from there it propagates to the level of the integral.

There is a note about the Euler-Lagrange equation (author: Preetum Nakkiran), in which the Euler-Lagrange equation is derived using differential reasoning only. That is: stating the integral is skipped altogether. That demonstrates that stating the integral is not necessary for deriving the Euler-Lagrange equation. https://preetum.nakkiran.org/lagrange.html

At the start of this comment I announced: the suggested contrast between traditional approach (force-acceleration) and Lagrangian approach is only an apparent contrast. On closer examination we see the two formalisms are in fact very closely connected.

I created demonstrations with interactive diagrams.

http://cleonis.nl/physics/phys256/calculus_variations.php The following case is used as motivation for developing Calculus of Variations: the shape of a soap film stretching between two coaxial rings. (The name of the solution is 'catenoid'; a surface of revolution.) Then the discussion moves to the Catenary problem: to calculate the shape of a hanging chain. The two problems have the same solution; the curve is the hyperbolic cosine.

Demonstration of Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...

The diagrams have sliders. Moving the sliders sweeps out variation of a trial trajectory. The diagram shows how the kinetic energy and the potential energy respond to sweeping out variation.

I have created a demonstration of Hamilton's stationary action with interactive diagrams, (supported with discussion of the mathematics that is involved).

Interestingly: it is possible to go in all forward steps from Newtonian mechanics to Hamilton's stationary action. That is the approach of this demonstration. (How Hamilton's stationary action came into the physics community is quite a convoluted story. With benefit of hindsight: a transparent exposition is possible.)

Recommended: read the following two articles in this order: Introduction to calculus of variations: http://cleonis.nl/physics/phys256/calculus_variations.php

Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...

The path from F=ma to Hamilton's stationary action goes in two stages: 1) Derivation of the work-energy theorem from F=ma 2) Demonstration that in cases where the work-energy theorem holds good Hamilton's stationary action will hold good also

Also interesting: Within the scope of Hamilton's stationary action there are also classes of cases such that the true trajectory corresponds to a maximum of Hamilton's action.

In the demonstration it is shown for which classes of cases the stationary point corresponds to a minimum of Hamilton's action, and for which classes to a maximum.

The point is: it is not about minimization. The actual criterion is that which both have in common: As you sweep out variation: in the variation space the true trajectory is the one with the property that the derivative of Hamilton's action is zero. The interactive diagrams illustrate why that property holds good (it follows from the work-energy theorem).

Hamilton's stationary action is a mathematical property. When the derivative of the kinetic energy matches the derivative of the potential energy: then the derivative of Hamilton's action is zero.

(Ycombinator does not give control over the layout of the text I submit. I insert end-of-line, to structure the text, but they are eaten.)

About Hamilton's stationary action (which you refer to as 'least action').

I have created an educational resource in which I address the question of how it comes about that F=ma can be recovered from Hamilton's stationary action. This resource gives a two-pronged approach: the concepts are illustrated with interactive diagrams, and parallel to that a full presentation of the mathematics.

I start with a discussion of the nature of Calculus of Variations. I use the problem of a soap film stretching between two parallel concentric rings as motivating example. This leads to a derivation of the Euler-Lagrange equation.

Then I move to the Catenary problem. Interestingly, with the catenary problem both approaches are possible; you can solve for the catenary with differential calculus (as Leibniz did) or you can apply calculus of variations. What that means is that the catenary problem can serve as a Rosetta stone, offering a bridge between differential calculus and calculus of variations.

http://cleonis.nl/physics/phys256/calculus_variations.php

The discussion specifically for Hamilton's stationary action is in an article of its own:

http://cleonis.nl/physics/phys256/energy_position_equation.p...

Here is one way of looking at it: statistical mechanics introduced the concept of entropy.

Years ago, in school, the physics teacher gave the following vivid demonstration:

The demonstration involved two beakers, stacked, the openings facing each other, initially a sheet of thin cardboard separated the two.

In the bottom beaker a quantity of Nitrogen dioxide gas had been had been added. The brown color of the gas was clearly visible. The top beaker was filled with plain air, so it was colorless.

Nitrogen dioxide is denser than air. If the gases would not mix then all of the Nitrogen dioxide would stay in the bottom beaker. But of course the two do mix.

When the separator was removed we saw the brown color of the Nitrogen dioxide rise to the top. In less than half a minute the combined space was an even brown color.

And then the teacher explained the significance: in the process of filling the entire space the heavier Nitrogen dioxide molecules had displaced lighter molecules. That is: a significant part of the population of Nitrogen dioxide had moved against the pull of gravity. This move against gravity is probability driven.

Statistical mechanics provides the means to treat this process quantitatively. You quantify by counting numbers of states. Mixed states outnumber separated states - by far.

The climbing of the Nitrogen dioxide molecules goes at the expense of the temperature of the combined gases. That is, if you make sure that in the initial state the temperature in the two compartments is the same then you can compare the final temperature with that. The temperature of the final mixture will be a bit lower than the starting temperature. That is, some kinetic energy has been converted to gravitational potential energy.

So in this particular demonstration probability was acting in a direction opposite to gravity, and overall probability had the upper hand.

Probability effects fall in the category of emergent phenomena. An emergent phenomenon is somewhat of an in-between category. Not quite as fundamental as the law of gravity, but there is no denying that it has an existence of its own.

Sure enough, the principles of Carnot's thermodynamics and the premises of statistical mechanisc look quite differently. The thing is: since both form the same thermodynamics there must be a connection.

I submit: the qualification 'completely independent axioms' is incorrect. There is the observation: temperature is transitive. This transitive property is a statement of conservation (In terms of Carnot's thermodynamic it used to be thought of as conservation of Caloric.) The concept of Conservation of a quantity correlates with information.

We have that statistical mechanics subsumed Carnot's thermodynamics.

The laws of Carnot's thermodynamics are theorems of statistical mechanics. (Those theorems weren't necessarily stated explicitly. I'm saying the principles of statistical mechanics are sufficient to imply the laws of Carnot's thermodynamics.)

Specifically about the Lagrangian of Classical Mechanics (Hamilton's Action) I have discussed that on physics.stackexchange https://physics.stackexchange.com/a/670705/ The ideas are expressed in diagrams. What is expressed in the diagrams is repeated/corroborated in mathematical expressions (stackexchange supports Mathjax). The visualizations leave no room to get lost.

It is in fact possible to explain Hamilton's action within the context of classical mechanics.

On physics.stackexchange I have discussed that, in an answer posted in oktober 2021. That discussion is illustrated with animated GIF's. The animated GIF's are composed of successive screenshots of interactive diagrams that are on my own website.

https://physics.stackexchange.com/a/670705/

Stackexchange has mathjax support, and support for uploading images, that is why I refer to my post on physics.stackexchange

The following is to give you an idea of what I discuss.

We have that if F=ma is granted as axiom then the Work-Energy theorem follows as theorem.

(As we know: the derivation of the Work-Energy theorem is subject to the following condition: it is only applicable if it is possible to define an unambiguous expression for potential energy. In order to have a well-defined expression for potential energy the force that is involved needs to be a conservative force.)

The Work-Energy theorem implies the following: In the process of interconversion of potential energy and kinetic energy: the rate of change of kinetic energy always matches the rate of change of potential energy. (If the potential energy is decreasing then the kinetic energy is increasing at the same rate)

In terms of exploring a variation space of trial trajectories:

The true trajectory has the following properties: A property of derivative with respect to time: - At every point along the trajectory the derivative of the kinetic energy with respect to time matches the derivative of the potential energy with respect to time.

A property of derivative with respect to _position_: - At every point along the trajectory the derivative of the kinetic energy with respect to _position_ matches the derivative of the potential energy with respect to _position_.

I want to highlight this: In mathematical models that describe changes taking place we are accustomed to taking the derivative with respect to _time_.

But: When we are representing the physics taking place in terms of _Energy_ it is powerful to take the derivative with respect to _position_.

In classical mechanics: When you insert the Lagrangian in the Euler-Lagrange equation then the operation that the Euler-Lagrange equation performs is that it takes the derivative of the Lagrangian with respect to position.

You are looking for the point where the _derivative_ of the Lagrangian with respect to _position_ is zero.

when that derivative is zero the derivative-of-the-kinetic-energy-with-respect-to-position _matches_ the derivitive-of-the-potential-energy-with-respect-to-position.

For the concept of stationary action minimum or maximum is immaterial. Stationary action is about identifying the point in variation space such that at every point along the trajectory the derivative-of-the-kinetic-energy-with-respect-to-position matches the derivitive-of-the-potential-energy-with-respect-to-position

Finally: There is a concept that I will refer to as Jacob's lemma. (This concept was introduced by Jacob Bernoulli in the course of presenting his solution to the Brachistochrone problem. The Brachistochrone problem had been presented by Johann Bernoulli, as a challenge.)

Jacob's Lemma was stated decades before Euler started development of calculus of variations. Jacob's Lemma is crucial to understanding calculus of variations.

Take the Brachistochrone curve. Divide it in subsections. Then each subsection is in and of itself an instance of the Brachistochrone problem. This process of subdivision can be repeated indefinitely. In the end you have a concatenation of infinitissimally short subsections, and we know that each of those subsections is an instance of the Brachistochrone problem.

This tells us that a differential equation must exist that solves the Brachistochrone problem. (It does not narrow down what that differential equation is, but at least you have logical proof that it _must_ exist.)

In fact, Jacob Bernoulli succeeded in solving the Brachistochrone problem using a differential calculus approach.

The Euler-Lagrange equation takes the variational formulation, and converts it to differential equation form.

I have created a resource that I think addresses your dissatisfaction.

The information is available on physics.stackexchange https://physics.stackexchange.com/a/670705

I use 'Hamilton's stationary action' to refer to the action concept of Classical Mechanics.

For Hamilton's stationary action the standard presentation is that it is demonstrated that F=ma can be recovered from Hamilton's stationary action.

Here's the thing: in physics it is common that derivation can be performed in either direction, and that applies in this case too. Hamilton's stationary action can be derived from F=ma

The derivation proceeds in two stages: 1. Derivation of the Work-Energy theorem from F=ma 2. Demonstration that in all cases where the Work-Energy theorem holds good Hamilton's stationary action will hold good also

Importantly, it's not retracing of the steps. The from-F=ma-to-Hamilton derivation hinges on the Work-Energy theorem. It's a different path altogether.

The steps of the derivation show why Hamilton's stationary action holds good. It achieves the justification you are looking for.

The derivation that I present is for the case of Hamilton's stationary action specifically; I'm positive the reasoning generalizes to all areas where an action concept is applied.

The demonstration is illustrated with interactive diagrams. (On physics.stackexchange the diagrams are posted as animated GIFs, the frames of the GIF are successive screenshots of the interactive diagram.)

Each diagram has one or more sliders, to explore variation of a trial trajectory. The diagram shows how the kinetic energy and potential energy respond to variation sweep.

The interactive diagrams are on my own website: http://cleonis.nl/physics/phys256/energy_position_equation.p...

I have created an exposition of Hamilton's stationary action that is visualization based. The visualizations consist of interactive diagrams. Each diagram represents a case where the visitor can sweep out a range of trial trajectories (using a slider), to home in on the true trajectory.

The three main diagrams have multiple sliders, allowing the visitor to make a local adjustment to the trial trajectory. The diagrams display how the kinetic energy and the potential energy respond to change of the trial trajectory.

http://www.cleonis.nl/physics/phys256/energy_position_equati...

The objective of the exposition is to make Hamilton's stationary action entirely transparent.

You mention stationary action.

Interestingly, F=ma and Hamilton's stationary action are mutually derivable.

The usual presentation is to show that Hamilton's stationary action implies the newtonian formulation. Interestingly, it is also possible to start with F=ma, and move in all forward steps to Hamilton's stationary action. No additional assumptions are required.

I created a series of interactive diagrams to make Hamilton's stationary action entirely transparent.

http://www.cleonis.nl/physics/phys256/energy_position_equati...

(Of course, quantum mechanics is the overall deeper theory; we assume that classical mechanics holds good because classical mechanics is a limiting case of quantum mechanics; at scales large enough that quantum effects average out the outcome in terms of quantum mechanics converges onto the outcome in terms of classical mechanics.)

I have a hard copy of the first edition. The dedication says: "dedicated to the principle of least action"

I have an educational resource for introduction to Hamilton's stationary action. The title is "Least action visualized".

http://www.cleonis.nl/physics/phys256/least_action.php

The diagrams on the page have a slider for active exploration. Moving the slider sweeps out a range of trial trajectories. As you change the trial trajectory the diagram shows how the graphs of the energies come out accordingly.

In this resource Hamilton's stationary action is introduced in a two-stage process.

First stage: We have the Work-Energy theorem, which we can apply with equal validity in infinitisimal form. The true trajectory has the following obvious property: at every instant in time the rate of change of potential energy matches the rate of change of kinetic energy. Demanding this match as a condition we identify the true trajectory among the range of trial trajectories. That is, this initial stage is already variational approach, but it doesn't yet use the concept of action.

Second stage: Demonstration of moving in a single step from the first stage to Hamilton's stationary action.

The demonstration is for the simplest case: a uniform force, hence a linear potential. The reasoning generalizes to all cases.