# Lagrangian without K.E and any anthromorophic answer

(Yes, I have searched the other posts and each one comes up deficient for what I want.)

Why must the Lagrangian be extremized? And why it is of the form L = T – V?

BUT I HAVE CAVEATS!

Please do it from first principles WITHOUT an understanding of F=ma.

(And, yes, I understand the calculus of variations.)

In other words, I have read, many times, that that form of the Lagrangian is SELECTED to ensure Newton’s law. I don’t like that “SELECTED” nonsense. I am looking for an intuitive understanding of why THAT form must be extremized.

The first answer here comes close:
http://www.quora.com/Laymans-Terms/What-is-a-Lagrangian [Broken]
But I do not like that answer because....
I do not want to hear any anthropomorphic explanation, e.g.: “The ball wants to…” or “The object desires…”

For then I would just ask you “WHY does the ball want to…”

So first please explain it with regard to K.E. and Potential E. just so I can understand that much...

And then, to make matters worse, I am hoping you can repeat the answer WITHOUT using the words KINETIC or POTENTIAL. (Because if you use those words, you are imbuing your response with a tacit awareness of F=ma)

So, instead, please use 0.5mv-squared and mgh (yes, you can focus on particle that has been thrown into the air: I am fine with that)

In other words,for a ball thrown into the air and without any reference to Lagrangian, Kinetic or Potential or Action

Why is the following extremized:
1. the mass weighted square of the velocity halved
2. the negative of the mass weighted height
If the constraints I placed on the possible answer are excessive, please tell me why. For it might be that such observational constraints are a necessary bridge between the intuition and the mathematical formalism. It may simply be that "the ball really DOES want to."

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Orodruin
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This is not a scientific question, just as asking why F = ma is not a scientific question. We use it because it is a very good model of how the world around us behaves and it has been thoroughly tested - as well tested as Newtonian mechanics, since they are equivalent. The upshot of reformulating mechanics is that the symmetries of a system become more obvious and that the step to quantum mechanics is less of a leap of faith.

This is not a scientific question, just as asking why F = ma is not a scientific question. We use it because it is a very good model of how the world around us behaves and it has been thoroughly tested - as well tested as Newtonian mechanics, since they are equivalent. The upshot of reformulating mechanics is that the symmetries of a system become more obvious and that the step to quantum mechanics is less of a leap of faith.
OK, so then no "explanation" in the traditional sense... how about just an interpretation along the lines of that link I posted.
Because that author comes really close to an interpretation. I just get muddled up on the "ball wants to" part.
And it would be great to hear that author's comment without reference to KE or Pot.Energy

Orodruin
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Why do you want to do it without referencing kinetic and potential energy? In Lagrangian mechanics, those are fundamental concepts and the forces are derived from these concepts, not the other way around.

Well, I am trying to say something like this... and I acknowledge I may be way off the mark.

If one throws a ball up with a force (not saying F=ma here... just force), the force maximizes the height.
(or minimizes the negative of the height).
But the height is weighted with the mass.

Second, there is velocity on the way up. One initially minimizes that, but find that the square of the velocity, weighted with the mass is more reasonable.
And then I get lost there.

I have this gut feeling it can be done... Or maybe I should accept your first answer and simply accept the "form" of the "model."

vanhees71
Gold Member
You can answer the question in some sense on a scientific basis. If you want to answer such "why questions", you have to argue starting from a more comprehensive theory than the theory under consideration. Within classical physics, Hamilton's principle (in Hamiltonian form) is the most fundamental basis from which to start. So it cannot be derived from somewhere but you can only argue that it was successful in describing classical physics (Newtonian and relativistic mechanics, electromagnetics).

The more comprehensive theory to start from is quantum theory, which you can formulate in terms of an action principle, known as the path-integral formalism or Schwinger's quantum action. The classical path is the one you obtain from the full path integral in the stationary-phase approximation, which holds if the typical action of the system under consideration is very large compared to ##\hbar##. The reason is that for paths which are not stationary in this sense, the integrand in the path integral is very rapidly oscillating, leading to a cancellation of such paths. For the stationary path the phases from paths close to it add up, and make the most important contribution to the integral. This explains why the propagation of the dynamical system is dominated by the solution of the equations of motion that extremize the action.

vanhees71
Gold Member
That looks like an introduction to one of Noether's theorems, and it's well worth studying it carefully. If I had to choose one result of pure math most important to (theoretical) physics then it's Noether's theorems. As the text states, indeed you have to guess a Lagrangian, but this guess must be guided by observations of nature, and the best way to translate from observations to math is (at least as seen from a science-historical perspective) to find the general symmetries of the problem and work them in systematically into the Lagrangian/Hamiltonian. So our guesses must be educated ones, and the most successful heuristic tool are the relations between symmetries (math) and conservation laws (empirically founded natural laws, i.e., physics).