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 Classical mechanics has been superseded by Quantum mechanics, with classical mechanics as limiting case; the validity of classical mechanics must be attributed to classical mechanics emerging from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.
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.
I haven't had a chance to really dig into your resource yet but I am definitely going to do so. Perhaps I'll wait a few days until the change you mentioned is implemented.
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.
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.
I will argue that 'has least action as foundation' does not in itself imply that Lagrangian mechanics is a sparser theory:
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.
You seem to know more about this than me, but it seems to me that the first law does more than just induce a metric, I've always thought of it as positing inertia as an axiom.
There's also more than one way to think about complexity. Newtownian mechanics in practice requires introducing forces everywhere, especially for more complex systems, to the point that it can feel a bit ad hoc. Lagrangian mechanics very often requires fewer such introductions and often results in descriptions with fewer equations and fewer terms. If you can explain the same phenomenon with fewer 'entities', then it feels very much like Occam's razor would favor that explanation to me.
Indeed inertia. Theory of motion consists of describing the properties of Inertia.
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.
Nicely said, but I think then we are in agreement that Newtownian mechanics has a bit of redundancy that can be removed by switching to a Lagrangian framework, no? I think that's a situation where Occam's razor can be applied very cleanly: if we can make the exact same predictions with a sparser model.
Now the other poster has argued that science consists of finding minumum complexity explanations of natural phenomena, and I just argued that the 'minimal complexity' part should be left out. Science is all about making good predictions (and explanations), Occam's razor is more like a guiding principle to help find them (a bit akin to shrinkage in ML) rather than a strict criterion that should be part of the definition. And my example to illustrate this was Newtonian mechanics, which in a complexity/Occam's sense should be superseded by Lagrangian, yet that's not how anyone views this in practice. People view Lagrangian mechanics as a useful calculation tool to make equivalent predictions, but nobody thinks of it as nullifying Newtownian mechanics, even though it should be preferred from Occam's perspective. Or, as you said, the physics content is the same, but the complexity of the description is not, so complexity does not factor into whether it's physics.
Hi, I want to respond to a post from you from 2019. (That 2019 thread no longer offers the reply button, otherwise I would reply there of course.) I apologize for using this thread to get my message in.
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.
If I don't hear back in a week or so I will remind you, I hope that's OK with you.
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.
Some feedback (apologies in advance for it being negative):
- I mean this kindly, but this is getting a tad bit aggressive. Please let me glance at my own pace (or not glance, if I don't find the interest to), rather than periodically following up with me around the forum to make sure I look at what you wrote. I was much more interested in this problem at the time I came across it than I was in 2019, and I was similarly more interested in 2019 than I am now. During this time I have both (a) come across other explanations that I've found ~semi-satisfactory, and (b) gathered other things to occupy my brain with. While I do get some enjoyment from the topic, refreshing my understanding of analytical mechanics is really not my topmost interest or priority right now. It could easily be months or years before I become interested in the problem again to even think about it.
- I find it rather... jarring to see a sentence like "Summing bars of signed area, in the limit of subdividing into infinitely many bars: that is evaluating the integral" in the middle of an explanation about Lagrangians and the principle of stationary action. Yes... integration is the limit of addition; you should hope your reader knows that well by now. You wouldn't re-teach "multiplication is really just repeated addition" in the middle of a lecture about Fourier series; this feels equally out-of-place. Not only does it make it seem like you don't know your audience, it also wastes the reader's time, and makes it hard for them to find any gems you might have shared.
- I took a quick look at your site and I don't actually see you attempt to explain the rationale for why the quantity of interest is T - V (as opposed to, say, T + V) anywhere. You mention "you are looking for the sweet spot: the point where as the object is moving the rate of change of kinetic energy matches the rate of change of potential energy", but... am I? Really? I certainly wasn't looking for that, nor have any idea why you thought I would be looking for that. It almost seems to assume what you were trying to prove!
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.
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.
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 have created a resource for the purpose of making Hamilton's stationary action transparent.
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.
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.
In retrospect: the earliest recognition of a conserved quantity was Kepler's law of areas. Isaac Newton later showed that Kepler's law of areas is a specific instance of a property that obtains for any central force, not just the (inverse square) law of gravity.
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)
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.
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.