Cleon Teunissen

Current main interest: Hamilton's stationary action. Explanation with interactive diagrams.

[Cleon Teunissen @cleontns]

Reasons why I regard the two recommended cases as more informative:
The case of quadratic potential is Hooke's law, which gives harmonic oscillation. Harmonic oscillation has multiple symmetries. Among them: period of oscillation is independent of amplitude.

As a consequence: take a half period, implement from displacement zero to next displacement zero. Then when you change the amplitude (from end point to end point) the value of the action remains the same. Action as a function of variation: not a minimum, not a maximum, not an inflection point: it's a flat line.

The case of potential increasing with the cube of the displacement:
Then at the point in variation space where the trial trajectory coincides with the true trajectory the value of the action is at a maximum.

That is: over the sequence of linear, quadratic, cubic we seen an inversion:
Linear potential: true trajectory at minimum
Quadratic: flat line
Cubic: true trajectory at maximum

[Cleon Teunissen @cleontns]

Recommendation: also implement the following cases:
- Potential increases with the square of the displacement
- Potential increases with the cube of the displacement

That forms an interesting sequence:
linear
quadratic
cubic

Quadratic: trajectory is sine curve.
Cubic potential: no analytic solution; find trajectory with numerical analysis.

In the discussion on my own website all three are implemented (linear, quadratic, cubic).
https://t.co/kmzyjqaSp8

In my opinion the cases of quadratic potential and cubic potential are way more informative than the case of linear potential.

[Cleon Teunissen @cleontns]

@kudo_akio The potential-increases-with-cube-of-displacement case has a single global extremum; that convergence criterion is met.

[Cleon Teunissen @cleontns]

The following may be of interest to you. I created an interactive diagram for the following case: acceleration when the potential is not linear, but it increases with the cube of the displacement: E_p = h^3
https://t.co/eSx815TjNM
Exploring higher power potential energy function is interesting.

Possibly it will be of interest to you to set up the cubic potential with your numerical action evaluator.

On my website:
The potential-is-cube-of-displacement case is part of a presentation of classical mechanics stationary action.
There is another implementation, with multiple sliders, for granular control. Of that design three instances are implemented:
- linear potential
- quadratic potential (Hooke's law)
- potential increases with cube of displacement

[Cleon Teunissen @cleontns]

About propagation of surface waves of water. It is possible to create (more or less) a wave beam. Let two of those beams be crossing each other. In the overlapping area the wave pattern is the sum of the two waves. Interestingly: _beyond_ the overlap area: each original wave.

That is: crossing of two surface wave beams is very close to being _lossless_. After exiting the overlap area we see that the original wave continuous as if along the way nothing has happened.

To be clear: the only thing that the beams do is that they make the losslessness visible to recognize. If both sources create expanding circle waves the superposition of oscillation in the overlap area is lossless also, but since there is hardly any area without overlap there is little to no opportunity to recognize the lossless character.

With sound: creating a fairly narrow beam is quite doable, for instance with a parabolic sound reflector. Let the two sound emitters produce a sine wave of equal frequency, with a wavelength of, say, 20 centimeter or so. You can have two beams of sound crossing; beyond the overlap volume you would not be able to tell the difference.

On the other hand, if you place a microphone _inside the overlap volume_, right at a point where the sound waves of the two sound emitters happen to be in counterphase, then the microphone will register very little sound.

Interaction with a microphone is change of medium. At change of medium something irreversible may happen. Destructive interference of sound waves is something that happens as a superposition of waves interacts in an _irreversible_ way.

Beyond the overlap volume: no loss. As long as the sound propagation remains in the same medium nothing irreversible happens.
The same stated in another way: as long as the propagation is just through air any superposition is (very nearly) lossless.

[Cleon Teunissen @cleontns]

Hello, I'm aware you haven't been active here on Twitter since september 2025; it could be this account is abandoned. But then maybe you still have a peek from time to time.

About Lagrangian mechanics:
I created a resource for classical mechanics stationary action.
https://t.co/kmzyjqaSp8
The mathematics is illustrated with interactive diagrams.

About the content of the resource:
For classical mechanics stationary action the usual presentation is to show that F=ma can be recovered from it.
Interestingly, it is also possible to go from F=ma to stationary action.
The process has two stages:
- derivation of the work-energy theorem from F=ma
- transformation: from the work-energy theorem to stationary action.

So we have: while the usual presentation is to go ahead with assumption of stationary action, assumption is not a necessity. It is possible to go from the work-energy theorem to stationary action in a series of steps, each step reversible.

This notification was triggered by your column about Lagrangian mechanics, in 'Spectrum der Wissenschaften', and 'Scientific American', in July of 2026.

[Cleon Teunissen @cleontns]

As we know, stationary action relates to the phenomenon of conserved quantities.

In the particular case of classical mechanics:
when the rate of change of kinetic energy matches the rate of change of potential energy the derivative of the action is zero.

I created a resource for classical mechanics stationary action:
https://t.co/kmzyjqaSp8

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

The remarkable thing is the recurring theme of conserved quantity.

As to stationary action: if there is a conserved quantity then mathematically there is a way to express that property in the form of a bespoke action.

[Cleon Teunissen @cleontns]

Een speerpunt voor mij is: er op attenderen dat het in klassieke mechanica gaat over *stationaire actie*.

De interpretatie in termen van geringste actie is wijdverbreid. Schrijvers van leerboeken geven aan: het gaat feitelijk over stationaire actie, maar omdat iedereen gewend is aan de naam 'geringste actie' gebruiken we die naam.

Het ware traject heeft de eigenschap: de afgeleide van de actie (naar toegepaste variatie), is nul.

Het punt is: er zijn ook categorieën van gevallen waarbij het ware traject samenvalt met een punt in de variatie-ruimte waar de actie een maximum bereikt.

Wat de diverse categorieën gemeenschappelijk hebben: de afgeleide van de actie (naar toegepaste variatie), is nul.

Datgene dat de diverse categorieën gemeenschappelijk hebben, dat is het deel dat telt.

[Cleon Teunissen @cleontns]

Mogelijk dat het volgende jou aanspreekt:
Ik heb een hulpbron voor klassieke mechanica stationaire actie ontwikkeld.
https://t.co/j0dlQNFACH
De wiskunde wordt geïllustreerd met interactieve diagrammen. Een schuifregelaar verplaatsen past variatie toe. Het diagram toont hoe de kinetische energy en de potentiële energie daarop reageren.

Het belangrijkste voorbeeld dat wordt behandeld (meerdere diagrammen), is voor het geval van een potentiaal die toeneemt met de derde macht van de verplaatsing.

Een dergelijke potentiaal zal in de fysieke wereld maar zelden voorkomen; ik heb dat geval gekozen omdat dat geval bij uitstek geschikt is voor het demonstreren van stationaire actie.

[Cleon Teunissen @cleontns]

I have continued adding and rewriting. In the past years a lot has changed. I expect revisiting will be worth it.

The standard presentation is to show that F=ma can be recovered from H's stationary action.

I go in the opposite direction. I start with F=ma, and after a series of steps I end up at Hamilton's stationary action.

The transformation process has two stages:
- Derivation of the work-energy theorem from F=ma
- Demonstration: when the work-energy theorem holds good Hamilton's stationary action holds good also.

That is to say: the relation between Hamilton's stationary action and F=ma is bi-directional.

I think many people are unaware of this bi-directionality.

[Cleon Teunissen @cleontns]

Possibly of interest to you: I created an educational resource for Hamilton's stationary action. The mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation, the diagrams show how kinetic/potential energy respond.

https://t.co/kmzyjqaSp8

It's part of a set of three articles.
The other two articles:

Fermat's stationary time
https://t.co/nqad2Ffhn7

The catenary, soap film, and calculus of variations
https://t.co/IWXJvyXplr

[Cleon Teunissen @cleontns]

@star_lings @QiaochuYuan I found the Chad Orzel blog post with the comment by Ted Chiang, it's from 2008.

https://t.co/K0tJheFG3z

[Cleon Teunissen @cleontns]

Some years ago I read a comment by Ted Chiang to a blog post by the physicist Chad Orzel. (The blog post discussed the plot device of the movie 'Arrival'.)

Ted Chiang mentioned that when he wrote 'Story of your life' he was aware that the apparent teleological interpreration of classical mechanics doesn't carry over to quantum physics. Specifically, Ted Chiang wrote that he was aware that the path integral formulation of quantum physics offers no room for teleological interpretation.

If memory serves me, Ted Chiang acknowledged: the premise of 'Story of your life' is from a physics point of view not a possible one.

For the story the apparent possibility of teleological interpretation (in classical mechanics) served as inspiration.

My opinion: the value of the story isn't dependent on whether the physics checks out, the value is in how the idea is developed in-universe.

So, in my opinion, not including indepth discussion of the stationary action concept was a good decision. Including it would expose the story to falsification.

[Cleon Teunissen @cleontns]

In retrospect we can see that it was an instance of jumping to a conclusion.

For any curve: a point where the derivative is zero is one of the following three: a minimum, a maximum, or a stationary point of inflection.

In the cases initially examined the equation was such that the stationary point was an extremum, and the equation could be rearranged such that the extremum was a minimum.

The jumping-to-a-conclusion was, in my opinion, an instance of confirmation bias. The scholars were hoping to find a minimum formulation, and to them it appeared they had found just that.

I have created a resource for concepts in variational approach in classical physics:

Fermat's stationary time:
https://t.co/nqad2Ffhn7

Soap film, catenary, and calculus of variations:
https://t.co/IWXJvyXplr

Classical Mechanics stationary action:
https://t.co/kmzyjqaSp8

[Cleon Teunissen @cleontns]

The following may be of interest to you.
I created a resource of classical mechanics stationary action. The mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation, see how kinetic energy and potential energy respond.
https://t.co/kmzyjqaSp8

Classical mechanics stationary action is like a machine with internal moving parts. The interactive diagrams offer a view onto the internal moving parts.

Mathematically the exposition proceeds as follows:

1) the work-energy theorem is derived from F=ma

2) Discussion of the relation between the work-energy theorem and classical mechanics stationary action: demonstration that when the work-energy theorem holds good then stationary action holds good also.

The resource as a whole consist of three articles:
The other two:
Optics: Fermat's stationary time
https://t.co/nqad2Ffhn7

Statics: soap film, catenary, and calculus of variations
https://t.co/IWXJvyXplr

[Cleon Teunissen @cleontns]

The following may be of interest to you.
I created a resource for the classical mechanics stationary action concept.
https://t.co/kmzyjqaSp8
The mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagram shows how kinetic energy and potential energy respond

In section 7.1 John Taylor writes:
"Notice first that the Lagrangian is the KE _minus_ the PE. [...] You are certainly entitled to ask why the quantity T-U should be of any interest. There seems no simple answer to this question except that it is, as we shall see directly."

I have to say: John Taylor is underestimating the possibilities here.
The combination of interactive diagrams, and derivation that is visualized with the diagrams, goes a long way to make the classical mechanics stationary action concept transparent.

[Cleon Teunissen @cleontns]

To avoid misunderstanding: I'm in no position to grasp the neurophysiological material.

Possibly my resource can be of service to you in the following way: when you need to explain to others just how different classical mechanics stationary action and the path integral formulation are you can refer to the exposition of classical mechanics stationary action.

Repeating the link:
https://t.co/kmzyjqaSp8

The overall resource consists of three articles:

Discussion of the nature of Fermat's stationary time
https://t.co/nqad2Ffhn7

Discussion of the nature of Calculus of Variations
https://t.co/IWXJvyXplr

[Cleon Teunissen @cleontns]

Some remarks about classical mechanics stationary action:
We have: the true trajectory has the property that the derivative of Hamilton's action wrt applied variation is zero.

This derivative-is-zero criterion expresses the constraint that the true trajectory has the property that the rate of change of kinetic energy matches the rate of change of potential energy.

Matching rate of change => derivative-of-action-is-zero

Depending on the circumstances the derivative-is-zero-point can be a minimum, a maximum, or an inflection point. In nature all three occur. (The is-a-minimum outcome is encountered more often than the other outcomes. Presumably that has been a factor in the name 'least action' getting adopted.)

In classical mechanics the name 'least action' is most unfortunate; it suggests something that isn't there.

Classical stationary action identifies the path such that everywhere along the path the rate of change of kinetic energy matches the rate of change of potential energy.

Referring to you mentioning 'local forces description':

Contrary to what some authors suggest there is no room for a supposed local-global dichotomy. The demand is that everywhere along the trajectory rate of change of E_k matches rate of change of E_p. The demand everywhere along the trajectory is in one go a local criterion and a global criterion.

[Cleon Teunissen @cleontns]

My interest in the Lohmiller and Slotine article (and the comment by Vattay), comes from my interest in the stationary action concept of classical mechanics.

It could be that Lohmiller and Slotine are under the impression that the stationary action concept of classical mechanics is analogous to the path integral formulation of quantum mechanics, to the extent of being able to use it as a replacement for the path integral formulation. However, that doesn't work out: classical mechanics stationary action doesn't contribute quantum properties. (Instead other aspects of the calculution introduce the required quantum properties)

I created an educational resource for the classical mechanics stationary action concept. The mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagrams show how the kinetic energy and the potential energy respond.

https://t.co/kmzyjqaSp8

classical mechanics stationary action is like a machine with internal moving parts. The interactive diagrams of the resource offer a window onto those internal moving parts.

I hope I can persuade you to check out the resource.

Cleon Teunissen