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(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?

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

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:

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:

- the mass weighted square of the velocity halved
- the negative of the mass weighted height

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