- #1
bronxman
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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?
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
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:
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
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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