Exploring Why Hamilton's Principle Uses L=KE-PE Lagrangian

[Trying2Learn]

TL;DR Summary

Proof

As I understand, Hamilton's Principle with the L = KE - PE Lagrangian, leads to Newton's equation.

Why MINUS and not PLUS?

I have seen many attempts at such a proof and I now realize that one does not prove PRINCIPLES. Like Laws, they are observations. It produces an equation that describes the physics -- simple as that.

I am wondering, however, if someone can point me to a specific statement to this effect. I would prefer to see it written down from someone in a way that is not facile or overly complex. (I do not trust myself.)

Even in his lecture notes, Feynman does not express the reason for L = KE - PE

And let me be clear with this edit: I AM NOT ASKING WHY THIS IS THE WAY IT IS. I am asking why NO established textbooks address the inevitable question of WHY SO MANY ANTICIPATE that it could be T+V?

 

[andresB]

It's just the way it is, it is the Lagrangian that gives the equation of motion. But, you can try the other way around (from Newton equations to D'Alembert principle to Hamilton principle), and see from where the minus sign comes from. See for example Lanczos's "the variational principles of dynamics".

 

[Trying2Learn]

It's just the way it is, it is the Lagrangian that gives the equation of motion. But, you can try the other way around (from Newton equations to D'Alembert principle to Hamilton principle), and see from where the minus sign comes from. See for example Lanczos's "the variational principles of dynamics".

So let me ask the question this way...

Feynman is brilliant. In his notes, he states the L = T - V. Why did he not anticipate that some people would question why T-V and not T+V. Surely, he should have recognized this. But he does not say a word about this. It seems like an important issue: one does not prove this, one accepts it as a law.

Does any document on this issue specifically call out this point?

I trust you, I trust myself. But I would like to see it stated clearly in the literature, and not only on this chat.

 

[bob012345]

When Lagrange developed his equations of motion based on generalized coordinates he used $T$ and $V$ and noticed that the difference occurred.

$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right ) = \frac {\partial T}{\partial q_j} - \frac {\partial V}{\partial q_j}$$

Since $V$ does not depend on $\dot{q}_j$, it was convenient to introduce $L = T - V$ and rewrite it in compact form as

$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j} $$

https://en.wikipedia.org/wiki/Lagrangian_mechanics
Fowles' Analytical Mechanics, 3rd edition chapter 9

 

[Trying2Learn]

When Lagrange developed his equations of motion based on generalized coordinates he used $T$ and $V$ and noticed that the difference occurred.

$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right ) = \frac {\partial T}{\partial q_j} - \frac {\partial V}{\partial q_j}$$

Since $V$ does not depend on $\dot{q}_j$, it was convenient to introduce $L = T - V$ and rewrite it in compact form as

$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j} $$

OK then...

Then why did Lagrange not specifically call it out, himself, to wit... "Hey, dudes! I'm freaked. I would have anticipated the sum of KE and PE, not the difference. Yet it produces the result, and it is a law and one does not prove laws."

Why does no one seem to call this point out? I mean, google on the net and so many people try to prove it, and their attempts are all flawed in some way. Yes, I accept that this is the way it is and I am not asking WHY is it this way, but why have so few people called this point out, specifically?

 

[bob012345]

OK then...

Then why did Lagrange not specifically call it out, himself, to wit... "Hey, dudes! I'm freaked. I would have anticipated the sum of KE and PE, not the difference. Yet it produces the result, and it is a law and one does not prove laws."

Why does no one seem to call this point out? I mean, google on the net and so many people try to prove it, and their attempts are all flawed in some way. Yes, I accept that this is the way it is and I am not asking WHY is it this way, but why have so few people called this point out, specifically?

Because he was not trying specifically to formulate the total energy of the system which is what the Hamiltonian is. See the derivation of Lagrange's equations in Fowles' Analytical Mechanics, 3rd edition chapter 9. The Lagrangian is a definition, not something one has to prove. Lagrange was dealing with generalized forces and force is the negative derivative of potential energy

$$F_i = - \frac{\partial V}{\partial x_i}$$

That is where the minus sign ultimately comes from.

 

[Trying2Learn]

Because he was not trying specifically to formulate the total energy of the system which is what the Hamiltonian is. See the derivation of Lagrange's equations in Fowles' Analytical Mechanics, 3rd edition chapter 9. The Lagrangian is a definition, not something one has to prove.

OK, so I am sorry, but I am going to push this...

"The Lagrangian is a definition, not something one has to prove."

Yes, I understand that. Totally clear. Yet, look at the internet. So many people try to put forth proofs and they are all facile.

I am not interested in all these nuances. My question is pointed and direct.

Why does NO esteemed physicist call it out directly and simply?

I only hear arguments like your's and all the other ones, above on this thread and they are all digressions from my question.

Why did no one seem to anticipate that this would cause consternation? Why do arguments like yours (and I agree with you and understand) only appear when someone questions this. Why do the physics books not call it out directly and simply and state: "you don't prove this. It is a law."

 

[PeroK]

I would have anticipated the sum of KE and PE, not the difference.

The sum is no good, as KE + PE is a constant in most systems. Looking at that is just looking at the conservation of total energy.

The Lagrangian concept is that nature tries to balance KE and PE by minimising the difference. It's giving both equal priority in some sense.

 

[bob012345]

OK, so I am sorry, but I am going to push this...

"The Lagrangian is a definition, not something one has to prove."

Yes, I understand that. Totally clear. Yet, look at the internet. So many people try to put forth proofs and they are all facile.

I am not interested in all these nuances. My question is pointed and direct.

Why does NO esteemed physicist call it out directly and simply?

I only hear arguments like your's and all the other ones, above on this thread and they are all digressions from my question.

Why did no one seem to anticipate that this would cause consternation? Why do arguments like yours (and I agree with you and understand) only appear when someone questions this. Why do the physics books not call it out directly and simply and state: "you don't prove this. It is a law."

Not a law, a definition out of convenience. Nothing more which is why people generally aren't up in arms about it. If potential energy or force were defined differently perhaps it would look different but the important point is everything would still be consistent.

 

[Trying2Learn]

Not a law, a definition out of convenient. Nothing more which is why people generally aren't up in arms about it. If potential energy or force were defined differently perhaps it would look different but the important point is everything would still be consistent.

Again, I know that. You said it. Many have said it. But when it is discussed in formal treatments or textbooks, no one anticipates this. Why does not a single author anticipate this. I don't know why my question is not clear. I am not searching for a reason. I am searching for why this issue -- and a casual glance on the net shows that MANY have this issue -- is not specifically addressed.

 

[bob012345]

Again, I know that. You said it. Many have said it. But when it is discussed in formal treatments or textbooks, no one anticipates this. Why does not a single author anticipate this. I don't know why my question is not clear. I am not searching for a reason. I am searching for why this issue -- and a casual glance on the net shows that MANY have this issue -- is not specifically addressed.

Why they don't anticipate confusion that readers will ask why isn't $L =T + V$ instead of $T - V$? Probably because it is a definition and most students are just not going to be that analytical by questioning definitions. I suspect what you see on the internet ultimately is greatly multiplied confusion from just one or a few sources making it look like an issue. We see misunderstanding going viral more often these days.

 

[Trying2Learn]

Why they don't anticipate confusion that readers will ask why isn't $L =T + V$ instead of $T - V$? Probably because it is a definition and most students are just not going to be that analytical by questioning definitions. I suspect what you see on the internet ultimately is greatly multiplied confusion from just one or a few sources making it look like an issue. We see misunderstanding going viral more often these days.

Hmm... Interesting. I have to think about this one. Thank you.

 

[Dale]

Why does NO esteemed physicist call it out directly and simply?

How can we answer a question like this? You would have to ask said esteemed physicists.

It never seemed like an issue that needs addressing to me.

 

[Vanadium 50]

You got your answer. It's a definition. Like many things in physics, we define quantities that turn out to be useful.

Action is useful. But action is not energy.