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.
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
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.
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.
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.
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.)
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.
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