# Principle of least action as a law?

• anorlunda
In summary: B. This should be clear.In summary, the laws of nature must be confirmed and checked for invariance, but many can be derived from the principle of least action. The principle of least action is not considered a law, but rather a foundational concept in modern physics. It has been subjected to confirmation and invariance tests, and is typically dependent on the chosen Lagrangian. The action is typically a single valued function, but there may be multiple sets of laws depending on the Lagrangian chosen. The distinction between laws and principles can be ambiguous, but both have their value in understanding the behavior of the world. Ohm's law, for example, is often seen as a law but it can also be an approximate
anorlunda
Staff Emeritus
We say that the laws of nature (e.g. Newton's Laws, relativity, ...) must be confirmed and must be checked for invariance (e.g Lorentz, gauge, ...). Yet many of these laws may be directly derived from the principle of least action.

Why do we not consider the principle of least action a law of nature; indeed the premiere law? It seems to be the foundation of so much else.

Has the principle of least action been subjected to confirmation and invariance tests similar to other laws?

Is least action a property of this universe, or of all universes?

Is the action necessarily a single valued function? If not, there could be multiple local minima of action, and thus multiple sets of laws.

We say that the laws of nature (e.g. Newton's Laws, relativity, ...) must be confirmed and must be checked for invariance (e.g Lorentz, gauge, ...). Yet many of these laws may be directly derived from the principle of least action.

Not exactly. Rather the rules have to be confirmed experimentally many times, before they acquire the status of a law. But then usually the resulting law is taken as basic assumption in theoretical physics; it is not possible to confirm it with 100 % certainty, or prove it.

The laws do not have to obey any invariance, such as you indicate. Rather the presence or lack of invariance in some situations is a law of nature; but there may be other laws that do not respect such invariance.

Why do we not consider the principle of least action a law of nature; indeed the premiere law? It seems to be the foundation of so much else.
We can consider it a concrete law of nature, if the Lagrangian is supplemented. Otherwise, as a general law, it seems too vague to be useful. I think of it rather as of a kind of mathematical language, or scheme. But nevertheless it is interesting that so much physics can be formulated within this language.

Has the principle of least action been subjected to confirmation and invariance tests similar to other laws?
Both depend on the Lagrangian chosen, for concrete situation. There are Lagrangians which work well for some situations, and usually they have some sort of invariance. There are many examples in mechanics, motion of particles in external EM field, ...

Is the action necessarily a single valued function? If not, there could be multiple local minima of action, and thus multiple sets of laws.

I think so, since it is integral of one valued Lagrangian.

anorlunda said:
Why do we not consider the principle of least action a law of nature; indeed the premiere law? It seems to be the foundation of so much else.
I think that modern physics has largely stopped looking for "laws" and has stopped labeling things as "laws". I don't think that has anything to do with the value of the principle of least action, it is merely a reflection on modern naming conventions, IMO. For instance, Newton's laws first and second laws can be taken as definitions rather than expressions about physics, and his third law is essentially a postulate. Ohm's law can be taken as a definition of resistance. Etc. So now we just talk about the definitions and postulates of a theory and don't bother labeling them "laws".

I think 'principles' are less prescriptive than laws.

Laws can generally be expressed as an equation or formula eg Boyles Law, principles are more by way of a procedure eg D'Alemberts Principle, St Venant's Principle and so on.

Distinguishing between definitions, theorems and laws can be more problematic.

For example a geodesic is defined as the shortest line between two points in a space and finding the geodesic is not called a law or even a principle.

Perhaps the only real truth is that the last law was called Murphy's Law.

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For instance, Newton's laws first and second laws can be taken as definitions rather than expressions about physics, and his third law is essentially a postulate. Ohm's law can be taken as a definition of resistance.

We should not play down successes of physics. Above mentioned laws were great discoveries in their time. Even today, stated in modern language and refined, they are valid statements about the behavior of the world; they are not definitions.

Newton's laws require lengthier discussion, but in case of Ohm's law, I would like to say something in defense of laws. Roughly said, Ohm's law says this:

For most metal wires, the current flowing through them, if low enough, is directly proportional to the applied voltage.

This can be formulated by the equation

I =U/R

where I is current, U voltage and constant R, independent of U or I, is called resistance of the wire.

Of course, when the voltage is too high, this law leads to substantial error in current. That is why Ohm's law is an approximate law.

Sometimes it is useful to define effective, current-dependent resistance ##R_{ef}## by the equation

$$R_{ef}(I) =\frac{U}{I},$$

even if ##R(I)## varies with ##I##. But this does not mean that Ohm's law is valid. Ohm's law is not the equation; it is the direct proportionality.

Jano L. said:
We should not play down successes of physics.
I don't know why you would think that any of my comments above were "playing down successes of physics". There is immense value in coming up with useful concepts and defining them clearly.

Jano L. said:
Above mentioned laws were great discoveries in their time. Even today, stated in modern language and refined, they are valid statements about the behavior of the world; they are not definitions.
Then please define resistance without reference to Ohm's law.

Jano L. said:
Ohm's law is not the equation; it is the direct proportionality.
I don't even know what you think this means. The equation is the direct proportionality between voltage and current. Ohm's law is the direct proportionality between voltage and current. Therefore the equation is Ohm's law. If A is B and C is B then A is C.

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Not sure if it helps, but Feynman in Lectures Vol 1 said of the least time of light:

"With Snell's theory we can "understand" light. Light goes along, it sees a surface, it bends because it does something at the surface. The idea of causality, that it goes from one point to another, and another, and so on, is easy to understand. But the principle of least time is a completely different philosophical principle about the way nature works. Instead of saying it is a causal thing, that when we do one thing something else happens, and so on, it says this: we set up the situation, and light decides which is the longer time, or the extreme one, and chooses that path."

Not sure, but I think he's pointing to the difference between the kind of principle that assumes a time causal development (like a differential equation) and a principle that seems to suggest that the mechanics of the situation act with respect to the geometric environment as a whole, so that the apparent principle is space causal...?

DaleSpam,
when you say that those laws can be understood as merely definitions, it sounds as if you are suggesting that they are pure mental constructs and are true axiomatically, independently of the actual behaviour of the world. That's like saying that those statements are axiomatically always true.
But they are not always true; every above law has its caveats.

Effective resistance of the wire is positive number R defined by

$$R_{ef} = U/I,$$

where I is the current, U the voltage. As the VA characteristic of wire is not straight exactly, R may depend on I (as well as on other things).

My point is that this equation is not Ohm's law. It is just a definition of R. There are other definitions, like differential resistance

$$R_d = \frac{dU}{dI}.$$

Ohm's law is stated in my previous post: it is about the linearity of the VA characteristic, and is quite accurate for metals. For semiconductors, it is not very accurate, although the equation

$$R_{ef} = U/I$$

is valid exactly.

Jano L. said:
when you say that those laws can be understood as merely definitions, it sounds as if you are suggesting that they are pure mental constructs and are true axiomatically, independently of the actual behaviour of the world. That's like saying that those statements are axiomatically always true.
I don't know what you have against definitions. Why would you say "merely" definitions? We need to have definitions for the terms that we use in physics. I don't think that I am minimizing the value of the laws, I think that you are minimizing the value of definitions.

Jano L. said:
Effective resistance of the wire is positive number R defined by

$$R_{ef} = U/I,$$
And here you have used Ohm's law to define resistance, exactly as I said above. Ohm's law is a definition of resistance.

Jano L. said:
Ohm's law is stated in my previous post: it is about the linearity of the VA characteristic
You are making a distinction without a difference. Whether you want to say "linearity of the VA characteristic" or "direct proportionality between voltage and current" either way is saying exactly the same thing as what the equation says.

You have tried to express Ohm's law in Engilsh a couple of times in a couple of different ways and each time you have done so you have come up with a sentence which is the same as the usual mathematical expression for Ohm's law and which is the same as the definition for resistance given above.

Ohm contributed two important things. One was the definition for resistance, $R=V/I$. The other was the observation that resistance is fairly constant over a wide range of currents, $dR/dI \approx 0$, for many objects, such as wires. I don't know why you seem upset that the rather meaningless label "Ohm's law" was attached to the first rather than the second.

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There seems to be the idea that there is some sort of hierarchy of scientific knowledge, with "law" at the top - and that this is what sent this thread down the "Ohm's Law" digression. This is simply not true. If there is any hierarchy at all, "theory" would be at the top, although many theories are not or are no longer called such - e.g. "germ theory of disease".

Most "laws" are empirical observations. The explanation came later.

It's strange that, in these 'enlightened' times, people still seem to be looking for (and they assume they exists) the sort of 'Laws' that would be laid down by some sort of designer or authority figure. The 'laws' of Science are much more pragmatically arrived at than that and they are all open to extension or modification when extreme or novel situations need to be explained.
Otoh, many of the posts we get are from people who are determined that they have found loopholes and errors in the established laws that Science has found to work very well as models. And that is often in the context of very common and well explained situations. It's not like the tax laws, which you can get round by describing the situation in a different way in order not to pay as much.

I apologize to anorlunda for the digression on Ohm's law. Probably it was not his intention to go into this.

Dalespam, I agree with you that useful definitions have great value. However, I cannot agree with you that the above definition of resistance is the same as Ohm's law.

The disagreement is largely in the meaning of words, so it is probably better not to go further into this here. I have found that the Wikipedia article on Ohm's law is quite nice - I recommend especially the first sections and the picture with four IV-curves. Also, E. Purcell's exposition in his book Electricity and Magnetism, sec 4.3, is quite good exposition of the Ohm law.

Jano L. said:
I cannot agree with you that the above definition of resistance is the same as Ohm's law.

The disagreement is largely in the meaning of words, so it is probably better not to go further into this here.
That is why I prefer to express Ohm's law in terms of equations where the meaning is unambiguous:

Ohm's law
I=V/R http://en.wikipedia.org/wiki/Ohm's_law
I=V/R http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmlaw.html
V=IR http://www.physics.uoguelph.ca/tutorials/ohm/Q.ohm.intro.html
V=IR http://www.hamuniverse.com/ohmslaw.html
V=IR http://www.grc.nasa.gov/WWW/k-12/Sample_Projects/Ohms_Law/ohmslaw.html
ΔV=IR http://www.physicsclassroom.com/class/circuits/u9l3c.cfm
E=RI http://www.angelfire.com/pa/baconbacon/page2.html

I have yet to find one that identifies Ohm's law with the expression $dR/dI \approx 0$ or anything even remotely similar.

DaleSpam said:
That is why I prefer to express Ohm's law in terms of equations where the meaning is unambiguous:

Ohm's law
. . . . . . .
I have yet to find one that identifies Ohm's law with the expression $dR/dI \approx 0$ or anything even remotely similar.

That list of links is of mixed quality. Even NASA manages to involve R with Ohm's law - which is just a circular argument.
R is a Definition: R=V/I
Ohm's law, in 'modern terms' just states that R (a ratio of two measurable quantities) is constant over a large range of currents for metals. I don't see how it can be looked upon as a basic law because it only applies to substances with a particular structure; it's more a rule of thumb - albeit a reliable one.
Putting it in the form " $dR/dI \approx 0$" may not be a common thing to do but you can hardly argue with it.

sophiecentaur said:
Even NASA manages to involve R with Ohm's law - which is just a circular argument.
R is a Definition: R=V/I
This is my point. That equation is the one referred to when people say "Ohm's law". That equation is also the definition of resistance. Therefore, Ohm's law is the definition of resistance.

sophiecentaur said:
Ohm's law, in 'modern terms' just states that R (a ratio of two measurable quantities) is constant over a large range of currents for metals. ...
Putting it in the form " $dR/dI \approx 0$" may not be a common thing to do but you can hardly argue with it.
So try to find a reference which uses that equation or an equivalent one as the mathematical expression of what they call "Ohm's law". I couldn't find one.

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I don't think I need to find a reference to justify differentiating a simple expression. That differential is just another valid way of describing a relationship.

But, I disagree that Ohm's law is a "definition" - it's a measured (and subsequently justified) relationship which metals exhibit. Ohm's law doesn't tell you R=V/I, it tells you that R is a constant for metals and those are two different issues. In a world where there were very few metals and many non-linear conducting materials, you could still find that, in their version of electrical theory, V/I was defined as a quantity R which might only be independent of I in a minority of conductors. This minority of substances would probably be notable by their 'Ohm's Law' behaviour.

Exactly the same thing would happen in a world where structural materials were all soggy rubber and 'plastic -like'. Engineers would use Young modulus (linear and non linear) but, without steel springs being common, Hooke's Law would not be the first thing that students would come across.

sophiecentaur said:
I don't think I need to find a reference to justify differentiating a simple expression.
Of course you don't need a reference to justify differentiating any expression. You need a reference to justify calling it "Ohm's Law".

sophiecentaur said:
Ohm's law doesn't tell you R=V/I, it tells you that R is a constant for metals and those are two different issues.
I recognize that they are two different issues, which is why I explicitly wrote down the mathematical equation representing each one. They are two distinct concepts with two very different equations. The fact is that every reference I have seen gives the label "Ohm's Law" to the equation V=IR, not to the equation dR/dI≈0.

And we're back onto the distraction.

Please get back on topic. I'd hate to have to lock this thread.

OK Guv, we'll behave ourselves.

And we're back onto the distraction.

Please get back on topic. I'd hate to have to lock this thread.

PS DaleSpam - I'll see you behind the bike sheds with rolled up umbrellas!

## 1. What is the principle of least action as a law?

The principle of least action is a fundamental law of physics that states that a physical system will always follow a path or trajectory that minimizes the total action, which is the integral of the Lagrangian over time. This law is used to describe the motion of particles and systems in classical mechanics.

## 2. How does the principle of least action relate to classical mechanics?

The principle of least action is a cornerstone of classical mechanics, along with Newton's laws of motion. It provides a more general and elegant way to describe the motion of particles and systems, and it is often used to derive the equations of motion for various physical systems.

## 3. Can the principle of least action be applied to other fields of science?

Yes, the principle of least action has also been applied to other areas of physics, such as quantum mechanics and electromagnetism. It has also been used in fields outside of physics, such as economics and biology.

## 4. What is the significance of the principle of least action in physics?

The principle of least action is significant because it provides a deeper understanding of the fundamental laws of physics and allows for more elegant and general descriptions of physical systems. It also helps to bridge the gap between classical and quantum mechanics.

## 5. Is the principle of least action a proven law of physics?

Yes, the principle of least action has been extensively tested and has been found to accurately describe the behavior of physical systems in a wide range of scenarios. It is considered to be a fundamental law of physics and has played a crucial role in our understanding of the natural world.

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