# Principle of Stationary Action - Intuition

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Dario56
Principle of stationary action allows us to find equations of motion if we plug appropriate lagrangian into Euler - Lagrange equation. In classical mechanics, this is the difference in kinetic and potential energy of the system.

However, how did Lagrange came to the idea that matter behaves this way? Without this idea, there would be no lagrangian mechanics.

Gold Member
Trying to avoid deja vu.

Have you tried reading his book?
https://www.amazon.com/dp/1418168432/?tag=pfamazon01-20
https://www.amazon.com/dp/1418167908/?tag=pfamazon01-20

It has been translated to English.
https://www.amazon.com/dp/0792343492/?tag=pfamazon01-20

The Preface of the translation has this

From the wiki in Post #2
"William Rowan Hamilton in 1834 and 1835[29] applied the variational principle to the classical Lagrangian function ... to obtain the Euler–Lagrange equations in their present form."

Taking the two comments together, I would GUESS that Hamilton actual created what we call the Lagrangian.

Edits:
from https://en.wikipedia.org/wiki/William_Rowan_Hamilton
"Paradoxically, the credit for discovering the quantity now called the Lagrangian and Lagrange's equations belongs to Hamilton."

The introduction to the translation provides some insights.

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Dario56
Trying to avoid deja vu.

Have you tried reading his book?
https://www.amazon.com/dp/1418168432/?tag=pfamazon01-20
https://www.amazon.com/dp/1418167908/?tag=pfamazon01-20

It has been translated to English.
https://www.amazon.com/dp/0792343492/?tag=pfamazon01-20

The Preface of the translation has this
View attachment 295650

From the wiki in Post #2
"William Rowan Hamilton in 1834 and 1835[29] applied the variational principle to the classical Lagrangian function ... to obtain the Euler–Lagrange equations in their present form."

Taking the two comments together, I would GUESS that Hamilton actual created what we call the Lagrangian.

Edits:
from https://en.wikipedia.org/wiki/William_Rowan_Hamilton
"Paradoxically, the credit for discovering the quantity now called the Lagrangian and Lagrange's equations belongs to Hamilton."

The introduction to the translation provides some insights.
Yes, but on what grounds physicists justified this principle at that time? How did they come up with this hypothesis? Without this idea, they would never plug lagrangian into Euler - Lagrange equation and thus confirm their hypothesis. Idea comes first and than Euler - Lagrange equation since that equation solves variational problem given that we want to make action stationary.

Gold Member
Physics is justified by the success of its predictions. In this case, we had Newton's laws and a belief in a principle of least action (where action is not clearly defined). Euler (early 18th century), Lagrange (late 18th century), Hamilton(early 19th century) and others successively refined and generalized the principle of least action until we have what we have today.

HomogenousCow
You can write any DE in terms of an extremization problem, it’s done in other fields without physical justification all the time.

Gold Member
2022 Award
From a modern point of view the action principle follows from QT. There are 2 approaches to see this. For me the most intuitive is Feynman's path integral to evaluate the propagator and evaluating it using the "method of steepest descent".

The other equivalent approach has been used already by Schrödinger to derive his "wave mechanics" in the backward direction. Given the Schrödinger equation, you get the classical trajectories in terms of the Hamilton-Jacobi PDE formulation by applying singular perturbation theory (aka the "eikonal approximation" to the Schrödinger equation, aka the "WKB approximation"). Schrödinger used the analogy between the relation of the naive photon picture a la Einstein, which is just geometrical optics, i.e., the eikonal approximation, of Maxwell's (wave) equations. So he asked for a wave equation whose wave equation leads to the Hamilton-Jacobi PDE and thus particle trajectories following the Newtonian equations of motion.

A masterful discussion of this can be found in Sommerfeld's Wave Mechanics (the famous addendum to the also famous "bible of atomic physics", "Atombau und Spektrallinien" ("Atomic Structure and Spectral Lines").

sysprog and PeroK
Gold Member
You had posted this question both on Reddit and Physicsforums, and initially I chose to respond on Reddit. However, on Reddit there is no provision to present equations.

On Reddit I had pointed out that the approach to mechanics introduced by Joseph-Louis Lagrange did not rely on calculus of variations to arrive at equations of motion.
What Lagrange did do was to make systematic use of expressing physics taking place in terms of interconversion of kinetic energy and potential energy.

That implies that Lagrange had a way of establishing how kinetic energy and potential energy are related, such that equations of motion can be obtained.

The following presents a way of establishing the relation, using means available at the time of Euler and Lagrange. I emphasize: these means were available at the time of Euler and Lagrange. The steps are not necessarily the same as how Lagrange proceeded. Rather, the following is a demonstration that it is possible.

When an object is going down a potential gradient, the work done by the force is moving the object from a state of higher potential to a state of lower potential.

Our starting point is to evaluate Work done by a force, evaluating from starting point ##s_0## to point ##s##
$$\int_{s_0}^s F \ ds \qquad (1)$$
The force causes acceleration: Newton's second law: ## F = ma ##
To find the effect of the force we substitute the ##F## in (1), and we will evaluate the integral of the acceleration.
For the time being I will not write the factor ##m##, it is a multiplicative factor that is just carried from step to step. In the final expression I will include the factor ##m## again.

In the next steps the following relations will be used:
$$ds = v \ dt \qquad (2)$$
$$dv = a \ dt \qquad (3)$$
The integral from a starting point s_0 to final point s
$$\int_{s_0}^s a \ ds \qquad (4)$$
Change of the differential according to (2), with corresponding change of limits:
$$\int_{t_0}^t a \ v \ dt \qquad (5)$$
Change the order:
$$\int_{t_0}^t v \ a \ dt \qquad (6)$$
Change of the differential according to (3), with corresponding change of limits:
$$\int_{v_0}^v v \ dv \qquad (7)$$
So we have:
$$\int_{s_0}^s a \ ds = \tfrac{1}{2} v^2 - \tfrac{1}{2} v_0^2 \qquad (8)$$
Including the factor ##m## again we see that we have arrived at the Work-Energy theorem:
$$\int_{s_0}^s F \ ds = \tfrac{1}{2} mv^2 - \tfrac{1}{2} mv_0^2 \qquad (9)$$

We have that the relation between F=ma and the Work-Energy theorem is as follows: start with F=ma and evaluate the integral with respect to position, both the left hand side and the right hand side. The result is the Work-Energy theorem.

The Work-Energy theorem has physics content and mathematics content.

Equation (8) states the mathematics content. (8) follows from the following relations:
velocity is the time derivative of position
acceleration is the time derivative of velocity

The physics content of the Work-Energy theorem is ##F=ma##.
That is: the physics content of the Work-Energy theorem does not extend beyond ##F=ma##; in terms of physics content the Work-Energy theorem and ##F=ma## are the same equation.

This derivation is quite general. The evaluation is in the form of integration so it is valid for cases where the force (hence the acceleration) changes as a function of position.

A logical consequence of the Work-Energy theorem is that when an object is moving down a potential gradient the kinetic energy will increase by the amount that the potential energy decreases. (With everything reversed when moving in the opposite direction, of course.)
$$\Delta E_k = -\Delta E_p \qquad (10)$$
(10) is valid down to infinitisimally short intervals of distance, so to find an equation of motion we can set up the following differential equation:
$$\frac{dE_k}{ds} = -\frac{dE_p}{ds} \qquad (11)$$
Of course, in theory of motion we are accustomed to differentiating with respect to time, so (11), which does differentiation with respect to position, looks peculiar.

The thing is: potential energy is the integral of force with respect to position, and kinetic energy is the integral of acceleration with respect to position. This means that when you differentiate with respect to position you recover F=ma.

Again, these steps are not necessarily the same as how Lagrange proceeded, but it shows that what Lagrange did in his 'Mechanique analytique' is possible. It is possible to establish the relation between kinetic energy and potential energy without using calculus of variations.

Last edited:
protonsarecool, sysprog, vanhees71 and 1 other person
Dario56
You had posted this question both on Reddit and Physicsforums, and initially I chose to respond on Reddit. However, on Reddit there is no provision to present equations.

On Reddit I had pointed out that the approach to mechanics introduced by Joseph-Louis Lagrange did not rely on calculus of variations to arrive at equations of motion.
What Lagrange did do was to make systematic use of expressing physics taking place in terms of interconversion of kinetic energy and potential energy.

That implies that Lagrange had a way of establishing how kinetic energy and potential energy are related, such that equations of motion can be obtained.

The following presents a way of establishing the relation, using means available at the time of Euler and Lagrange. I emphasize: these means were available at the time of Euler and Lagrange. The steps are not necessarily the same as how Lagrange proceeded. Rather, the following is a demonstration that it is possible.

When an object is going down a potential gradient, the work done by the force is moving the object from a state of higher potential to a state of lower potential.

Our starting point is to evaluate Work done by a force, evaluating from starting point ##s_0## to point ##s##
$$\int_{s_0}^s F \ ds \qquad (1)$$
The force causes acceleration: Newton's second law: ## F = ma ##
To find the effect of the force we substitute the ##F## in (1), and we will evaluate the integral of the acceleration.
For the time being I will not write the factor ##m##, it is a multiplicative factor that is just carried from step to step. In the final expression I will include the factor ##m## again.

In the next steps the following relations will be used:
$$ds = v \ dt \qquad (2)$$
$$dv = a \ dt \qquad (3)$$
The integral from a starting point s_0 to final point s
$$\int_{s_0}^s a \ ds \qquad (4)$$
Change of the differential according to (2), with corresponding change of limits:
$$\int_{t_0}^t a \ v \ dt \qquad (5)$$
Change the order:
$$\int_{t_0}^t v \ a \ dt \qquad (6)$$
Change of the differential according to (3), with corresponding change of limits:
$$\int_{v_0}^v v \ dv \qquad (7)$$
So we have:
$$\int_{s_0}^s a \ ds = \tfrac{1}{2} v^2 - \tfrac{1}{2} v_0^2 \qquad (8)$$
Including the factor ##m## again we see that we have arrived at the Work-Energy theorem:
$$\int_{s_0}^s F \ ds = \tfrac{1}{2} mv^2 - \tfrac{1}{2} mv_0^2 \qquad (9)$$

We have that the relation between F=ma and the Work-Energy theorem is as follows: start with F=ma and evaluate the integral with respect to position, both the left hand side and the right hand side. The result is the Work-Energy theorem.

The Work-Energy theorem has physics content and mathematics content.

Equation (8) states the mathematics content. (8) follows from the following relations:
velocity is the time derivative of position
acceleration is the time derivative of velocity

The physics content of the Work-Energy theorem is ##F=ma##.
That is: the physics content of the Work-Energy theorem does not extend beyond ##F=ma##; in terms of physics content the Work-Energy theorem and ##F=ma## are the same equation.

This derivation is quite general. The evaluation is in the form of integration so it is valid for cases where the force (hence the acceleration) changes as a function of position.

A logical consequence of the Work-Energy theorem is that when an object is moving down a potential gradient the kinetic energy will increase by the amount that the potential energy decreases. (With everything reversed when moving in the opposite direction, of course.)
$$\Delta E_k = -\Delta E_p \qquad (10)$$
(10) is valid down to infinitisimally short intervals of distance, so to find an equation of motion we can set up the following differential equation:
$$\frac{dE_k}{ds} = -\frac{dE_p}{ds} \qquad (11)$$
Of course, in theory of motion we are accustomed to differentiating with respect to time, so (11), which does differentiation with respect to position, looks peculiar.

The thing is: potential energy is the integral of force with respect to position, and kinetic energy is the integral of acceleration with respect to position. This means that when you differentiate with respect to position you recover F=ma.

Again, these steps are not necessarily the same as how Lagrange proceeded, but it shows that what Lagrange did in his 'Mechanique analytique' is possible. It is possible to establish the relation between kinetic energy and potential energy without using calculus of variations.
I am little late to reply as I was otherwise engaged. What you said has a lot of sense. Lagrange thought in terms of energy conservation. However, this doesn't explain how did physicists after Lagrange got to the idea of stationary action. As I said to solve the problem of what path matter takes, idea of stationary action comes first and than we justify it when Newton's 2nd law pops out of Euler - Lagrange equation.

However, unlike Newton's 2nd law, stationary action principle isn't easily observed in experiment (not to devaluate Newton's work), it is kind of abstract principle because of which it must come as an idea. This idea is than, as I said, justified when solving variational problem.

Question is: How did physicists come up with this idea?

Dario56
Physics is justified by the success of its predictions. In this case, we had Newton's laws and a belief in a principle of least action (where action is not clearly defined). Euler (early 18th century), Lagrange (late 18th century), Hamilton(early 19th century) and others successively refined and generalized the principle of least action until we have what we have today.
I agree. However, this doesn't explain how did physicists after Lagrange got to the idea of stationary action. As I said to solve the problem of what path matter takes, idea of stationary action comes first and than we justify it when Newton's 2nd law pops out of Euler - Lagrange equation.

However, unlike Newton's 2nd law, stationary action principle isn't easily observed in experiment (not to devaluate Newton's work), it is kind of abstract principle because of which it must come as an idea. This idea is than, as I said, justified when solving variational problem.

Question is: How did physicists come up with this idea?

Gold Member

On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function
(Dublin University Review (1833), pp. 795-826)

Hamilton said:
The former of these two laws [Law of Stationary Action] was discovered in the following manner. The elementary principle of straight rays shewed that light, under the most simple and usual circumstances, employs the direct, and, therefore, the shortest course to pass from one point to another. Again, it was a very early discovery, (attributed by Laplace to Ptolemy,) that in the case of a plane mirror, the bent line formed by the incident and reflected rays is shorter than any other bent line, having the same extremities, and having its point of bending on the mirror. These facts were thought by some to be instances and results of the simplicity and economy of nature; and Fermat, whose researches on maxima and minima are claimed by the continental mathematicians as the germ of the differential calculus, sought anxiously to trace some similar economy in the more complex case of refraction. He believed that by a metaphysical or cosmological necessity, arising from the simplicity of the universe, light always takes the course which it can traverse in the shortest time. To reconcile this metaphysical opinion with the law of refraction, discovered experimentally by Snellius, Fermat was led to suppose that the two lengths, or indices, which Snellius had measured on the incident ray prolonged and on the refracted ray, and had observed to have one common projection on a refracting plane, are inversely proportional to the two successive velocities of the light before and after refraction, and therefore that the velocity of light is diminished on entering those denser media in which it is observed to approach the perpendicular: for Fermat believed that the time of propagation of light along a line bent by refraction was represented by the sum of the two products, of the incident portion multiplied by the index of the first medium, and of the refracted portion multiplied by the index of the second medium; because he found, by his mathematical method, that this sum was less, in the case of a plane refractor, than if light went by any other than its actual path from one given point to another; and because he perceived that the supposition of a velocity inversely as the index, reconciled his mathematical discovery of the minimum of the foregoing sum with his cosmological principle of least time. Des Cartes attacked Fermat’s opinions respecting light, but Leibnitz zealously defended them; and Huygens was led, by reasonings of a very different kind, to adopt Fermat’s conclusions of a velocity inversely as the index, and of a minimum time of propagation of light, in passing from one given point to another through an ordinary refracting plane. Newton, however, by his theory of emission and attraction, was led to conclude that the velocity of light was directly, not inversely, as the index, and that it was increased instead of being diminished on entering a denser medium; a result incompatible with the theorem of shortest time in refraction. The theorem of shortest time was accordingly abandoned by many, and among the rest by Maupertuis, who, however, proposed in its stead, as a new cosmological principle, that celebrated law of least action which has since acquired so high a rank in mathematical physics, by the improvements of Euler and Lagrange. Maupertuis gave the name of action to the product of space and velocity, or rather to the sum of all such products for the various elements of any motion; conceiving that the more space has been traversed and the less time it has been traversed in, the more action may be considered to have been expended: and by combining this idea of action with Newton’s estimate of the velocity of light, as increased by a denser medium, and as proportional to the refracting index, and with Fermat’s mathematical theorem of the minimum sum of the products of paths and indices in ordinary refraction at a plane, he concluded that the course chosen by light corresponded always to the least possible action, though not always to the least possible time. He proposed this view as reconciling physical and metaphysical principles, which the results of Newton had seemed to put in opposition to each other; and he soon proceeded to extend his law of least action to the phenomena of the shock of bodies. Euler, attached to Maupertuis, and pleased with these novel results, employed his own great mathematical powers to prove that the law of least action extends to all the curves described by points under the influence of central forces; or, to speak more precisely, that if any such curve be compared with any other curve between the same extremities, which differs from it indefinitely little in shape and in position, and may be imagined to be described by a neighbouring point with the same law of velocity, and if we give the name of action to the integral of the product of the velocity and an element of a curve, the difference of the two neighbouring values of this action will be indefinitely less than the greatest linear distance (itself indefinitely small) between the two near curves; a theorem which I think may be advantageously expressed by saying that the action is stationary.
Lagrange extended this theorem of Euler to the motion of a system of points or bodies which act in any manner on each other; the action being in this case the sum of the masses by the foregoing integrals. Laplace has also extended the use of the principle in optics, by applying it to the refraction of crystals; and has pointed out an analogous principle in mechanics, for all imaginable connexions between force and velocity. But although the law of least action has thus attained a rank among the highest theorems of physics, yet its pretensions to a cosmological necessity, on the ground of economy in the universe, are now generally rejected. And the rejection appears just, for this, among other reasons, that the quantity pretended to be economised is in fact often lavishly expended. In optics, for example, though the sum of the incident and reflected portions of the path of light, in a single ordinary reflexion at a plane, is always the shortest of any, yet in reflexion at a curved mirror this economy is often violated. If an eye be placed in the interior but not at the centre of a reflecting hollow sphere, it may see itself reflected in two opposite points, of which one indeed is the nearest to it, but the other on the contrary is the furthest; so that of the two different paths of light, corresponding to these two opposite points, the one indeed is the shortest, but the other is the longest of any. In mathematical language, the integral called action, instead of being always a minimum, is often a maximum; and often it is neither the one nor the other: though it has always a certain stationary property, of a kind which has been already alluded to, and which will soon be more fully explained. We cannot, therefore, suppose the economy of this quantity to have been designed in the divine idea of the universe: though a simplicity of some high kind may be believed to be included in that idea. And though we may retain the name of action to denote the stationary integral to which it has become appropriated—which we may do without adopting either the metaphysical or (in optics) the physical opinions that first suggested the name—yet we ought not (I think) to retain the epithet least: but rather to adopt the alteration proposed above, and to speak, in mechanics and in optics, of the Law of Stationary Action.

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Omega0, andresB, protonsarecool and 1 other person
Gold Member
However, unlike Newton's 2nd law, stationary action principle isn't easily observed in experiment (not to devaluate Newton's work), it is kind of abstract principle because of which it must come as an idea.
I agree that without a stimulus an idea of stationary action is unlikely to be explored.

I get the impression that you are extremely reluctant to accept that Fermat's least time was that stimulus.

Anyway: in my opinion the way that Hamilton's stationary action is commonly presented is unnecessarily opaque. The student is presented with a lot of abstractions, and as you write: at the end F=ma pops out. But the presentation is unsatisfactory; presumably there is a reason why F=ma can be recovered from Hamilton's stationary action, but no indication is given what that reason is.

Hamilton's stationary action has internal moving parts, and I have created a series of animated diagrams to show all of that. You can think of those diagrams as exploded view drawings

This series of diagrams takes F=ma as starting point, and then it works in all forward steps to Hamilton's stationary action. That demonstration is available here on Physicsforums See: Hamilton's stationary action

(The animated diagrams are composited from screenshots of interactive diagrams that are on my own website.)

Stationary action versus least action
I notice that you use the name 'stationary action'. Indeed: the name 'least action' has to be abandoned.
It is interesting to see that Hamilton was already aware of that. Hamilton advocated the name 'Law of Stationary Action'.
(Yet in later times people relapsed to using the name 'least action. In my opinion that is an example of erosion of physics understanding.)

Hamilton points out that both in optics and in mechanics there are classes of cases where the true trajectory correponds to a maximum of the action.

We have the following logical principle: a single counter-example is already sufficient to refute a conjecture.
Depending on the circumstances of the case it may be that the true trajectory correspons to a minimum of Hamilton's action, or to a maximum of Hamiltion's action. Therefore the conjecture that the true trajectory always corresponds to a minimum of Hamilton's action is refuted.

This means that yet another level of abstraction must be admitted. One should not think of the process as trying to find a minimum or a maximum. Minimum or maximum is not relevant.

In the case of Hamilton's stationary action the only thing that is relevant is that the rate of change of kinetic energy must match the rate of change of potential energy. Hamilton's action is stationary - along the trial trajectory - if and only if the rate of change of kinetic energy matches the rate of change of potential energy.

I repeat the link to the demonstration available here on physicsforums: Hamilton's stationary action

Dario56
I agree that without a stimulus an idea of stationary action is unlikely to be explored.

I get the impression that you are extremely reluctant to accept that Fermat's least time was that stimulus.

Anyway: in my opinion the way that Hamilton's stationary action is commonly presented is unnecessarily opaque. The student is presented with a lot of abstractions, and as you write: at the end F=ma pops out. But the presentation is unsatisfactory; presumably there is a reason why F=ma can be recovered from Hamilton's stationary action, but no indication is given what that reason is.

Hamilton's stationary action has internal moving parts, and I have created a series of animated diagrams to show all of that. You can think of those diagrams as exploded view drawings

This series of diagrams takes F=ma as starting point, and then it works in all forward steps to Hamilton's stationary action. That demonstration is available here on Physicsforums See: Hamilton's stationary action

(The animated diagrams are composited from screenshots of interactive diagrams that are on my own website.)

Stationary action versus least action
I notice that you use the name 'stationary action'. Indeed: the name 'least action' has to be abandoned.
It is interesting to see that Hamilton was already aware of that. Hamilton advocated the name 'Law of Stationary Action'.
(Yet in later times people relapsed to using the name 'least action. In my opinion that is an example of erosion of physics understanding.)

Hamilton points out that both in optics and in mechanics there are classes of cases where the true trajectory correponds to a maximum of the action.

We have the following logical principle: a single counter-example is already sufficient to refute a conjecture.
Depending on the circumstances of the case it may be that the true trajectory correspons to a minimum of Hamilton's action, or to a maximum of Hamiltion's action. Therefore the conjecture that the true trajectory always corresponds to a minimum of Hamilton's action is refuted.

This means that yet another level of abstraction must be admitted. One should not think of the process as trying to find a minimum or a maximum. Minimum or maximum is not relevant.

In the case of Hamilton's stationary action the only thing that is relevant is that the rate of change of kinetic energy must match the rate of change of potential energy. Hamilton's action is stationary - along the trial trajectory - if and only if the rate of change of kinetic energy matches the rate of change of potential energy.

I repeat the link to the demonstration available here on physicsforums: Hamilton's stationary action
I agree that Hamilton's principle is opaquely presented and that I still didn't find answer to my question which I think is quite basic. I also agree that I am not completely satisfied with Fermat's principle giving idea to the stationary action since action and time aren't really very similar physical quantities. I do think that Fermat's principle gave physicists an idea to think about path of matter as a solution to some variational problem (as path of light is also solution to variational problem). That "some" is crucial since if you figure it out you know what path that is. Which brings us back to my question: How did abstract quantity like action came up as an idea?

Also, I make an important claim here. I claim that this principle must come FIRST (as an idea) in order for us to justify it. This is because we justify stationary action principle as a solution of variational problem when Newton's 2nd laws pops out of Euler - Lagrange equation.

In this regard, there is no talking about Euler - Lagrange equation if we don't know what is needed to be stationary.

Maybe physicists worked backwards. Maybe they thought, if matter takes path which makes some quantity stationary (like light does) than this path should satisfy Euler - Lagrange equation. First use Newton's 2nd law to define equation of motion. Than solve Euler - Lagrange equation inversely, find what is that quantity under derivative (which we know call lagrangian), but I am not sure if this method can be used to find lagrangian nor if it was actually used since I didn't find this anywhere.

weirdoguy and PeroK
In this regard, there is no talking about Euler - Lagrange equation if we don't know what is needed to be stationary.
On the contrary, one can indeed work backwards from EoM to find a suitable action. This is called the Inverse Problem in Lagrangian Mechanics. A bit of googling yields a large amount of literature on this subject.

If the techniques used therein fail to yield a suitable action, they sometimes give at least a hint (by studying the associated Helmholtz conditions, iirc) of what needs to be modified to get a satisfactory action.

protonsarecool, Dario56, vanhees71 and 1 other person