Visualization of Hamilton's stationary action with interactive diagrams

[Cleonis]

Hello,

my name is Cleon Teunissen,

I noticed your material about Hamilton's stationary action.

The following may be of interest to you:
I created a resource for Hamilton's stationary action in which the mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagrams shows the response.

To give you an idea: I have inserted a screenshot of the most comprehensive diagram of that resource; the screenshot is at the end of this post.
It's a 4 panel diagram, with a master slider below. When the master slider is shifted the representations in the four panels shift accordingly.

Hamilton's stationary action:
https://cleonis.nl/physics/phys256/energy_position_equation.php

The Hamilton's stationary action article is part of a set of three, the other two articles:
Fermat's stationary time:
https://cleonis.nl/physics/phys256/fermat_time.php

Calculus of variations, the catenary:
https://cleonis.nl/physics/phys256/calculus_variations.php

The article about Hamilton's stationary action opens with derivation of the work-energy theorem from F=ma
The work-energy theorem provides the transformation between representation in terms of force-acceleration and representation in terms of potentialEnergy-kineticEnergy

From there it is demonstrated:
When the circumstances are such that the work-energy theorem holds good then Hamilton's stationary action holds good also.

In the article about calculus of variations:
Among the things that are discussed there is the following observation:
The solution space of a differential equation is a space of functions.
(By contrast, for an equation that finds, say, the roots of a polynomial, the solution space is a space of numbers.)
There is another equation category with the property that the solution space is a space of functions: variational equation.

Overall: the relation between differential equations and variational equations is closer than generally appreciated. I find that relation very interesting.

The screenshot:

[40tude]

I added a link to your article in the webliography section. Can you add a link to my article on your post and let me know when this is done? Regards // ------------------- ​------------------------​ Le ven. 27 mars 2026, 08:51, Cleonis ***@***.***> a écrit :

…

Hello, my name is Cleon Teunissen, I noticed your material about Hamilton's stationary action. The following may be of interest to you: I created a resource for Hamilton's stationary action in which the mathematics is illustrated with interactive diagrams. Move sliders to sweep out variation. The diagrams shows the response. To give you an idea: I have inserted a screenshot of the most comprehensive diagram of that resource; the screenshot is at the end of this post. It's a 4 panel diagram, with a master slider below. When the master slider is shifted the representations in the four panels shift accordingly. Hamilton's stationary action: https://cleonis.nl/physics/phys256/energy_position_equation.php The Hamilton's stationary action article is part of a set of three, the other two articles: Fermat's stationary time: https://cleonis.nl/physics/phys256/fermat_time.php Calculus of variations, the catenary: https://cleonis.nl/physics/phys256/calculus_variations.php The article about Hamilton's stationary action opens with derivation of the work-energy theorem from F=ma The work-energy theorem provides the transformation between representation in terms of force-acceleration and representation in terms of potentialEnergy-kineticEnergy From there it is demonstrated: When the circumstances are such that the work-energy theorem holds good then Hamilton's stationary action holds good also. In the article about calculus of variations: Among the things that are discussed there is the following observation: The solution space of a differential equation is a space of functions. (By contrast, for an equation that finds, say, the roots of a polynomial, the solution space is a space of numbers.) There is another equation category with the property that the solution space is a space of functions: variational equation. Overall: the relation between differential equations and variational equations is closer than generally appreciated. I find that relation very interesting. The screenshot: screenshot__H_action.png (view on web) <https://github.com/user-attachments/assets/1ba25338-be63-4480-a4f4-b0795ed18858> — Reply to this email directly, view it on GitHub <#11?email_source=notifications&email_token=AALONDN3X4LAX3CTYP5X64D4SYXG5A5CNFSNUABAM5UWIORPF5TWS5BNNB2WEL2ENFZWG5LTONUW63RPHE3TMNZTGI3KM4TFMFZW63VKON2WE43DOJUWEZLEUVSXMZLOOSWGM33PORSXEX3DNRUWG2Y>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AALONDMGUHVMHOPHOBCJLW34SYXG5AVCNFSM6AAAAACXBR7DCKVHI2DSMVQWIX3LMV43ERDJONRXK43TNFXW4OZZG43DOMZSGY> . You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[Cleonis]

First thing to take care of: I apologize for my delay in responding.

I very much appreciate that in the webliography section you have added a link to my article .

Let me start with two quotes from you that I think are particularly relevant.

The key idea is that extremizing a functional is not a completely new kind of law. It is actually another way of encoding differential equations.

we are requiring that small changes of the function do not affect the quantity at first order. This constraint turns out to be strong enough to produce local equations of motion (through the Euler–Lagrange equations).

I'm emphasizing the above ideas because the resource that I created is structured along those lines.

In physics derivations can often be run in both directions.

Example: The Lagrangian formulation of mechanics is transformed to the Hamiltonian formulation by way of Legendre transformation. Legendre transformation is its own inverse: applying Legendre transformation twice recovers the original function.

We have:
F=ma
the work-energy theorem
Hamilton's stationary action

If F=ma is granted as axiom then the work-energy theorem follows as theorem.

As we know: when the force that is involved is a conservative force then work-energy theorem holds good.
In the resource it is demonstrated:
When the work-energy theorem holds good: Hamilton's stationary action holds good also.

That is: it is not just asserted that Hamilton's stationary action is equivalent to the work-energy theorem, the equivalence is demonstrated.

Also demonstrated: to arrive at the Lagrangian formulation of mechanics the work-energy theorem is sufficient.

Noether's theorem

We have the observation: for physical interaction (with exchange of momentum) there is a connection between symmetry of the interaction and a counterpart conserved quantity.
As we know, the significance of that was regognized by physicists when Noether's theorem was formulated.

Observation: the thrust of Noether's theorem is independent of which formulation of dynamics is being used.

An example of that is the derivation of conservation of angular momentum in terms of the newtonian formulation of mechanics.
https://cleonis.nl/physics/phys256/angular_momentum.php
In the Principia Newton derived a generalized version of Kepler's law of areas. As we know: the area law anticipated the concept of conservation of angular momentum.

Newton's derivation of the area law uses multiple elements, but there is one element that is unique to it: the area law obtains when the force that is acting is a central force; the area law obtains when the way the force acts has rotational symmetry.

My takeaway from that is:
The content of Noether's theorem is universal, in the sense that it is not dependent on how the physics theory is formulated.

It is customary to express Noether's theorem in terms of the Lagrangian formulation of mechanics, but that is not a necessity; for any formulation of mechanics: the symmetry-conservedQuantity connection will obtain.

As we know, for each field of physics the Lagrangian for that field of physics is a bespoke Lagrangian.
That bespoke Lagrangian encodes the properties that are specific to that field of physics.
When variation is applied the equations of motion for that field of physics are derived from that bespoke Lagrangian.

In classical mechanics the criterion:
derivative-with-respect-to-variation-of-the-action-is-zero
is equivalent to the criterion:
there is - continuously - conservation of energy

In the case of Maxwell's equations:
I think it is very much worthwhile to structure the process of deriving Maxwell's equations from the Lagrangian of electromagnetism in such a way that parallels with classical mechanics are highlighted.

To obtain the potentials: start with the equation of motion, and perform integration (integration wrt position coordinate)

To recover the equation of motion: differentiate wrt position coordinate.

(What has to be negotiated, of course, is that the magnetic potential is proportional to particle velocity. Remarkably, it's still possible to apply the criterion: energy is conserved. As we know, the Lorentz force acts perpendicular to the velocity, which of course means that in the case of an isolated charged particle the Lorentz force doesn't have opportunity to do work.)

Regards,

Cleon Teunissen