Significance of the Lagrangian

  • #1
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Main Question or Discussion Point

While I understand the use of the Lagrangian in Hamilton's principle, I have the gut feeling that there is more to it than meets the eye.
For instance, while the hamiltonian is conceptually easy to understand and even I could have thought about it, the Lagrangian is something else. I would never have thought about subtracting the potential energy from the kinetic energy. How was this found? was it just by accident? Did a monkey erase a plus sign in the Hamiltonian and put a minus? or were there some physical reasons that justified attempting to use the difference of T and V as opposite to their sum?. Or maybe someone Lagrange? Hamilton? was kind of bored and decided to have some fun by trying something different?
The way the subject is usually presented more or less along these lines:
Let there be a function which we call Lagrangian (L) defined by L=T-V. If we do this and that with this function, we obtain some very useful results.
It appears to me that the expression for the Lagrangian is so simple, that there should be some simple explanation of it's significance, which we could understand even before we start writing any equations.
If such an explanation exists, and you know it, I'll appreciate your sharing it with us.
-Alex-
 
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Answers and Replies

  • #2
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The Lagrangian is a concept that comes from the variational principle. When you put this quantity into a functional and you calculate the extremal value (derivative equals zero)you get newton's equations of motion.
On a more intuitive note : one can say that when you calculate the minimal action (this is the lagrangian put into an integral over all possible paths between two points) needed to go from one point to another, you get a motion which is described by the newton-equations.

Or the newtonian equations state that nature is as lazy as possible...That is why nature will allways aim for the situation with lowest possible potential energy.

regards
marlon
 
  • #3
ZapperZ
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To add to what Marlon has said, the Lagrangian/Hamiltonian mechanics arose out of the Least Action Principle. This is a different approach to the dynamics of a system than Newtonian mechanics that uses forces. Such approach, using the calculus of variation, is what produces this formulation, and even Fermat's least time principle.

http://www.eftaylor.com/leastaction.html

Zz.
 
  • #4
Galileo
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alexepascual said:
The way the subject is usually presented more or less along these lines:
Let there be a function which we call Lagrangian (L) defined by L=T-V. If we do this and that with this function, we obtain some very useful results.
Yeah, I had (have) the same problem. It stems from the principle of least action.
I've tried to find a book which explains it well, but they are hard to find.
Here's a quote from one of the books:

"In almost all textbooks, even the best, this principle is presented so that it is impossible to understand. (K. Jacobi, Lectures on Dynamics, 1842-1843) I have chosen not to break with tradition.
 
  • #5
ZapperZ
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Galileo said:
Yeah, I had (have) the same problem. It stems from the principle of least action.
I've tried to find a book which explains it well, but they are hard to find.
Here's a quote from one of the books:
Check the link I gave earlier. It has at least one link that gives an almost "trivial" derivation of the Lagrangian.

I strongly suggest that one covers calculus of variation to fully understand the principle of least action. I've mentioned Mary Boas's text in a few postings on here. She has a very good coverage of this and sufficient for most physics majors.

Zz.
 
  • #6
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ZapperZ:
I briefly looked through the links at E.F.Taylor's site and only saw one article that might provide a derivation of the Lagrangian. But I would have to read the article to make sure the derivation gives enough intuitive insight.
Also, in another article ( I think by E.F. Taylor) he talks about reducing the principle of least action to a differential form by bringing the starting and end points very close together. This might provide further insight. Thanks for your advice, I think it'll be very useful.

Marlon:
I understand your explanation and that is the explanation that I have found in the books. But it is not very satisfactory to me because it starts with the use of T-V instead of having T-V come out as the quantity derived.
With respect to your explanation of nature aiming for the lowest potential energy, I doubt this is correct. As a mater of fact, the least action principle minimizes the difference between kinetic and potential energy, which could be achieved by having the highest potential energy possible.
Also, I think the idea that Nature would try to economize some quantities by choosing the minimum (view which was supported by Maupertui) was kind of discredited when it was found that Nature was not aiming for a minimum of these quantities but an extremum, meaning it could as well be a maximum.
Thanks for your input Marlon. I hope you post again if you don't agree with what I just said.

Galileo:
I wonder why K.Jacobi chose not to break with tradition. Maybe it was too much work to look for an easy-to-understand explanation.
I have been taking a look at the book "The Variational principles of mechanics" by Cornelius Lanczos. Some of it is too advanced for me, but it has some sections that are quite enlightening. Specifically, he has a Chapter on D'Alembert's Principle and in the following chapter, it appears that he derives the Lagrangian from D'Alembert's principle (pgs.111-113) I would have to read it a couple times and think about it in order to understand it. If you can get a hold of a copy of the book I suggest you take a look at it.
 
  • #7
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hamiltonian is not always V + T, that occurs for example if time don't appears in the lagrangian
 
  • #8
arildno
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alex:
I will offer an argument which possibly yields a bit of insight on the (history of) "action" concept; however, this is my representation, and should not be regarded as authorative in any way:

1. The "vis vitae"-concept:
In 18'th century-physics, the quantity [tex]V_{s}=mv^{2}[/tex] (that is, twice the kinetic energy "T") was called the "vis vitae" (life force) of the physical system.
(I believe it was Leibniz who championed the concept)
2.Energy and action:
Note that if we combine the "hamiltonian" (T+V=E) with the Lagrangian, we gain for the "action" (A=T-V):
[tex]A=V_{s}-E[/tex]
Hence, a rough characterizetion of "action" is:
Action is "excess life force"; nature tends to minimize this

NB!
I have no references to support this view, one really should make a study of the evolution of physics in the 17-18th to find the "rationale" physicists at that time made of "least-action"

As of today, one might regard the "least-action-principle" as a mathematical trick, but it probably goes "deeper" than that.
 
  • #9
pervect
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MiGUi said:
hamiltonian is not always V + T, that occurs for example if time don't appears in the lagrangian
One of the examples I'm aware of where H is not equal to V+T is the restricted three body problem where we use rotating coordinates (or any problem that uses rotating coordinates for that matter).

[edit #2 total re-write]

We can write the inertial coordinates in terms of the rotating coordinates
[tex]
x_{inertial}=
x \left( t \right) \cos \left( \omega\,t \right) -y \left( t \right)
\sin \left( \omega\,t \right) \hspace{.5 in}
y_{inertial}=y \left( t \right) \cos \left( \omega\,t \right) +x \left( t \right)
\sin \left( \omega\,t \right)
[/tex]

We can then say that
[tex]
T = (\frac{d x_{inertial}}{dt})^2+(\frac{d y_{inertial}}{dt})^2
[/tex]
[tex]
T = 1/2\,m{{\it xdot}}^{2}+1/2\,m{{\it ydot}}^{2}+1/2\,m{x}^{2}{\omega}^{2}+1/2\,m{y}^{2}{\omega}
^{2}-m{\it xdot}\,y\omega+mx\omega\,{\it ydot}
[/tex]

and [tex] L = T - V(x,y) [/tex]

We can generate the energy function as follows
[tex]
h = xdot \frac {\partial L}{\partial xdot} + ydot \frac {\partial L}{\partial ydot} - L
[/tex]
[tex]
h = 1/2\,m{{\it xdot}}^{2}+1/2\,m{{\it ydot}}^{2}-1/2\,m{x}^{2}{\omega}^{2
}-1/2\,m{y}^{2}{\omega}^{2}+V \left( x,y \right)
[/tex]

Note that the energy function, which is the Hamiltonian before we make the variable substitution that changes xdot and ydot into px and py, is NOT equal to the energy of the system. This quantity -2*h, using the above variables, is often called the Jacobi intergal of the three body problem.

http://scienceworld.wolfram.com/physics/JacobiIntegral.html

We complete the transformation to the Hamiltonian in the usual variables by setting

[tex]
px = \frac {\partial L}{\partial xdot}\hspace{.5 in} py = \frac {\partial L}{\partial ydot}
[/tex]
[tex]
H = \frac {px^2}{2m} + \frac {py^2}{2m} + \omega (px \; y - py \; x)
[/tex]

We can compare H to the value of the kinetic energy in the same variables and again see it's not the same

[tex]
E = \frac {px^2}{2m} + \frac {py^2}{2m} + V(x,y)
[/tex]
 
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  • #10
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Arildno:
Thanks for your post. I was already somewhat familiar with the "viz vitae" (also known as "viz viva"). But as far as I can see, this quantity would be equivalent to the kinetic energy, except for a factor of two. I understand that in certain problems it may be more convenient by not requiring the division by two, but I think both quantities would be mostly interchangeable (after correcting for the factor 2).
With respect to the equations you post, I don't see the Lagrangian coming out of them. With respect to "Action" my understanding is that it represents the integration of the lagrangian with respect to time. I think the following would be the correct equations: (which don't explain my question either)
H=T+V
L=T-V
Vs=2T
A= Integral{L dt}
A= Integral{(T-V)dt}
If we were to consider only the case where total energy is conserved, then we can consider:
V=H-T
L=T-(H-T)
L=T-H+T
L=2T-H
L=Vs-H
A=integral{(Vs-H)dt}
But these last equations and the inclusion of the viz viva don't appear to throw any more light on the subject.
Something interesting is that if L=2T-H , then when considering alternative paths with the same energy, minimizing A would be equivalent to minimizing the integral of T with respect to time. But I guess in Hamilton's principle we have the freedom to choose paths with different total energy, which would make this a mute point.
 
  • #11
arildno
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Hmm..you're probably right.
So much for pet theories..
 
  • #12
turin
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alexepascual said:
... he talks about reducing the principle of least action to a differential form by bringing the starting and end points very close together.
IMO, this is crucial to a physical interpretation of the principle of least action. Otherwize, the principle seems kind of "spooky" (i.e. non-causal).




alexepascual said:
... I think the idea that Nature would try to economize some quantities by choosing the minimum ... was kind of discredited when it was found that Nature was not aiming for a minimum of these quantities but an extremum, meaning it could as well be a maximum.
Don't you think that may be a bit picky? Whether a relative extremum is specifically a maximum or a minimum depends on the convention imposed. However, you are neglecting yet a third possibility for the action of a physical path: inflection (or saddle-point). The length of the physical path must be stationary (according to variations of parameters about that path), but not necessarily an extremum.
 
  • #13
krab
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alexepascual said:
...the hamiltonian is conceptually easy to understand and even I could have thought about it...
Are you saying that you could have intuited that the total energy written as a function of space and momentum coordinates has the characteristic that partial derivatives w.r.t. the momentum coordinates give the time derivatives of the corresponding positions and partial derivatives with respect to the space coordinates are equal to the negative of time derivatives of the corresponding momenta? If so, I find it very hard to believe.
 
  • #14
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Turin:
Your observation is very interesting. I didn't think about inflection points. Woundn't this support my point that Hamilton's principle does not represent an attempt by Nature to obtain an economy in a certain quantity?.
I have read that Maupertui tried to give to the principle of "least action" (Maupertuisian action, not the Hamiltonian Action) a kind of magical meaning, as if some kind of intelligence had economy as a purpose.
Don't you think that the fact that Action does not per force need to be a minimum talks against Maupertui's interpretation?.
Do you think I am wrong in saying that Maupertui's interpretation has been Discredited?.
Do you agree with Marlon's statement (which I was arguing against?) Or do you have a different objection to it?
IMO, this is crucial to a physical interpretation of the principle of least action. Otherwize, the principle seems kind of "spooky" (i.e. non-causal).
The fact that the principle of least action can be proven equivalent to Newton's second law I guess would take some of the spookyness out of it.
But I agree that if we don't have a good intuitive understanding of how it translates to a causal approach, then it would still feel "spooky".
I have made some progress reading Cornelius Lanczos book. I still have to read more and re-read some sections to fully understand it.

Krab,
I am not saying that I could have come up with Hamiltonian mechanics myself. My statement was not an attempt to brag about my capacity. I was just trying to say that the Hamiltonian as the sum of kinetic and potential energy was sufficiently simple for someone like me to understand, as opposed to the idea of the Lagrangian.
It is a concern for me though, what the mental process that leads to discovery is. I think very often a concept that appears "magical", which we think we would have never been able to find, would appear less so if we knew the mental path the discoverer took.
 
  • #15
turin
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alexepascual said:
Your observation is very interesting. I didn't think about inflection points. Woundn't this support my point that Hamilton's principle does not represent an attempt by Nature to obtain an economy in a certain quantity?
I don't know to what extent you intend to take this analogy/personification. I have no doubt in my own mind that the principle has a profound meaning regarding physical reality, and there does seem to be some kind of tendancy to, dare I say, "mimimization," but it is probably better to refer to the phenomenon as "equilibration." A system "seeks" a state from which all deviations present the same variation in action, to first order. Of course, there seems to be this unwritten rule in physics that the dynamics are only unambiguous up to second order, which I consider also a rather obscure concept to try to get ahold of.




alexepascual said:
I have read that Maupertui tried to give to the principle of "least action" (Maupertuisian action, not the Hamiltonian Action) a kind of magical meaning, as if some kind of intelligence had economy as a purpose.
Don't you think that the fact that Action does not per force need to be a minimum talks against Maupertui's interpretation?
I don't know anything about that. It sounds like metaphysics to me (and I say that in condescention).




alexepascual said:
Do you think I am wrong in saying that Maupertui's interpretation has been Discredited?.
Do you agree with Marlon's statement (which I was arguing against?) Or do you have a different objection to it?
I think I basically agree with your position. I don't think that there is some underlying drive towards an extremum condition. Though, I also don't take any integral nearly as seriously as a good, solid derivative in physics. Integrals introduce extra ambiguity whereas derivatives eliminate them (up to a point).




alexepascual said:
The fact that the principle of least action can be proven equivalent to Newton's second law I guess would take some of the spookyness out of it.
But I agree that if we don't have a good intuitive understanding of how it translates to a causal approach, then it would still feel "spooky".
I argue that neither Newton's laws (obviously) nor the principle of least action fundamentally characterize physical behavior; however, to me the principle of least action seems more fundamental than Newton's laws, when considered infinitesimally (integration over a trivially small temporal range).
 
  • #16
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Turin,
I am quite frustrated because I had just typed a response to your post and it suddenly dissapeared and the editor window appeared blank again.
I'll try to reproduce my answer in condensed form.
I don't know to what extent you intend to take this analogy/personification
Actually it is not my analogy/personification but Maupertui's and all I have said is that it has been discredited, part of the reason being that it is metaphysical, and partly because if "Nature" (some resemblance of "God" here?) really had a "purpose", this purpose would not be one of "economy" as proposed by Maupertui, but one of "equilibrization" as you say.
So, it looks like we agree more than it first appeared.
I also agree that an understanding of the principle would have to be more in terms of a derivative rather than an integration over time. (Athough there seems to be a need to integrate at a point, wich results in the principle as conventionally stated). This is explained by Cornelius Lanczos, but I don't fully grasp it yet. His explanation uses the concept of "forces of inertia" where every time a particle is acclerated, the "impressed force" is opposed (and often cancelled) by this "force of inertia" (ma). But the forces of inertia would not cancel the impressed forces when there is a constraint that has not been eliminated by a change of coordinates. If I am not explaining this correctly, it is because I am still in the process of understanding it. There is a principle in connection with these "forces of inertia" wich is known as "D'alembert's principle".
With respect to the ambiguity above second order you mention, I am not familiar with that. It would be nice to have Eye_in_the_sky here. I am sure he would have some opinion about that.
 
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  • #17
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alexepascual said:
There is a principle in connection with these "forces of inertia" wich is known as "D'alembert's principle".
It's a shame that we don't see more discussion about this principle here. Its an interesting topic.

Pete
 
  • #18
pervect
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alexepascual said:
There is a principle in connection with these "forces of inertia" wich is known as "D'alembert's principle".
Well, as I understand it, to get to D'alembert's principle, you start with the principle of virtual work.

The way I look at it is that if you have a system in equilibrium, no work is being done on the system. In a metaphysical sense, it's "not moving", though this is not necessarily true in a literal sense. (This may be oversimplified, but it works for me)

If you exclude systems where the forces of constraint do any work (usually this excludes dissipative forces of constraint, I.e friction), you can say that the applied physical forces do no work at equilibrium. This is the principle of Virtual work.

Mathematically, we write:

[tex]
\sum F^{applied}_{i} \cdot \delta r_i = 0
[/tex]

D'alembert's principle starts off with this principle, but extends it to cover systems that are not in equilibrium.

To accomplish this we must do something rather clever. We take the equations for a non-equilibrium system, F = dp/dt, and re-write them as F - dp/dt = 0. We then reinterpret this equation to observe that if we physically applied additional forces dp/dt to the system, we would have a system that was in equilibrium. Now we can then apply the equations of virtual work, since our new system is at equilibrium


In equation form, we write

[tex]
\sum ( F^{applied}_i - \dot p_i) \cdot \delta r_i = 0
[/tex]

This is known as D'alembert's principle, and it allows us to proceed with the derivation of the Lagrangian. The next step in the derivation is to get rid of the physical coordinates ri through substitution and replace them with the generalized coordinates qi

However, I'll leave this to you and your textbook at this point.
 
  • #19
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Pervect:
Thanks for your nice introduction to D'Alembert's principle. I'll print it out and use it as a guide while I read Goldstein's explanation.
 
  • #20
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pervect said:
However, I'll leave this to you and your textbook at this point.
I know the principle and did this out several times in the last 20 years. I was simply saying that its an interesting topic that should be discussed more.

Pete
 
  • #21
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alexepascual said:
ZapperZ:
Marlon:
I understand your explanation and that is the explanation that I have found in the books. But it is not very satisfactory to me because it starts with the use of T-V instead of having T-V come out as the quantity derived.
With respect to your explanation of nature aiming for the lowest potential energy, I doubt this is correct. As a mater of fact, the least action principle minimizes the difference between kinetic and potential energy, which could be achieved by having the highest potential energy possible.
Also, I think the idea that Nature would try to economize some quantities by choosing the minimum (view which was supported by Maupertui) was kind of discredited when it was found that Nature was not aiming for a minimum of these quantities but an extremum, meaning it could as well be a maximum.
Thanks for your input Marlon. I hope you post again if you don't agree with what I just said.

.
Why do you think that when drops of oil fall into water, they always form circular surfaces. Ever heard of the isoperimetric-property and it's relation to the potential energy ?

The derivative of the functional associated with the lagrangian is minimized over all possible paths. I mean the variational principle basically looks for THE path along which the ACTION is minimal, not the Lagrangian...

Minimal action corrsponds to the minimal amount of work necessary to follow the selected path. Trust me nature is as lazy as possible. This is a common view on nature that physics must describe (i am not saying it has to or can explain it...)

Your point about why T-V is correct, it was not really my intention to solve this interesting riddle as to why exactly T-V.

regards
marlon
 
  • #22
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Marlon,
As far as I know, and as I have said earlier, I don't think the "principle of least action" (also known as Hamilton's principle) really implies that action will tend to be a minimum when compared to the adjacent virtual paths. I think I have read that "principle of least action" is a misnomer and that it should actually be called "principle of stationary action". I had also read some place that action was supposed to be an extremum, but as Turin has pointed out, we have to also consider inflection points, making it more correct to talk about stationary action.
If you ask me for examples where Action is a maximum or a stationary value rather than a minimum, I have none that comes to mind right now. But I think I have seen some examples in the past. Maybe someone else on this thread can help us on this.
 
  • #23
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alexepascual said:
Marlon,
As far as I know, and as I have said earlier, I don't think the "principle of least action" (also known as Hamilton's principle) really implies that action will tend to be a minimum when compared to the adjacent virtual paths. I think I have read that "principle of least action" is a misnomer and that it should actually be called "principle of stationary action". I had also read some place that action was supposed to be an extremum, but as Turin has pointed out, we have to also consider inflection points, making it more correct to talk about stationary action.
If you ask me for examples where Action is a maximum or a stationary value rather than a minimum, I have none that comes to mind right now. But I think I have seen some examples in the past. Maybe someone else on this thread can help us on this.

ok, then my view may be wrong. I always believed (i mean i thought) that least action really meant that the work necessary to go from a to b on a certain path had to be MINIMAL.

¨Perhaps someone else can correct me on this...


regards
marlon
 
  • #24
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marlon said:
ok, then my view may be wrong. I always believed (i mean i thought) that least action really meant that the work necessary to go from a to b on a certain path had to be MINIMAL.

¨Perhaps someone else can correct me on this...


regards
marlon
I've never known or heard of this principle meaning that the work was minimal. Its often called the principle of least action because the action is a minimum between two closely space points on the curve used in the action integral.

Pete
 
  • #25
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But i am confused right now. Shouldn't there be some kind of intuitive physical meaning as to why the newtonian equations of motion are the nice result of calculating the extremum of the action-functional??? I know the mathematical background of all this, but what about the logical physical explanation.

That's why I thought (apparently i am wrong) that work must be minimal, because every process in nature wants to evolve to that state of lowest potential energy (the equilibrium)


any comments on that ???

regards
marlon
 

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