Cleonis

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

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.

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.

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.

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.

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.

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.

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 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.

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.