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 Classical mechanics has been superseded by Quantum mechanics, with classical mechanics as limiting case; the validity of classical mechanics must be attributed to classical mechanics emerging from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.

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.

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.)

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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'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.

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.

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.)

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...

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.

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.)