Alternatives to the Lagrangian?

[Feynstein100]

I'm just getting started on Lagrangian mechanics and what I can't understand is, how did Lagrange discover the Lagrangian? Did he just randomly decide to see what would happen if we calculate KE - PE or T - V and then discovered that the quantity is actually mathematically and physically significant?
And if so, how do we know that some other combination of T & V isn't also mathematically significant? There are infinite possibilities. T*V, T/V, Tlog(V), T^V and so on. Idk it seems a bit arbitrary and random to me.

 

[Frabjous]

He didn’t.

I believe Hamilton did later.

 

[Feynstein100]

He didn’t.

View attachment 331734
I believe Hamilton did later.

Thanks for the correction but that still doesn't answer my question

 

[apostolosdt]

Thanks for the correction but that still doesn't answer my question

Look into Goldstein (2nd ed.), Section 1-4, on D'Alembert's Principle, as the starting point for getting the Lagrangian.

 

[Frabjous]

Thanks for the correction but that still doesn't answer my question

It does. He found a relation between T and V. He wasn‘t looking at f(T,V).

 

[Feynstein100]

It does. He found a relation between T and V. He wasn‘t looking at f(T,V).

No, I got that. I meant, how do we know that some other f(T,V) isn't significant too?

 

[Frabjous]

Lagrangian’s are not unique. You can multiply them by a constant or add terms $\frac {dF} {dt}$ for any function F(q,t). A magnetic field does no work on a particle, yet you can write down a lagrangian for a particle in a magnetic field.

edited

 

[topsquark]

No, I got that. I meant, how do we know that some other f(T,V) isn't significant too?

Are you asking, "L = T - V is important but how do we know that the "Feynsteinian" ( ) $F = T^2 + 4 V^3 \sqrt{E - 4}$ isn't just as important?"

The answer is that we don't know. Perhaps there is something else out there that we don't yet know about. But that's for the future. If it's out there, I'm sure that someone will (eventually) come across it at some point in time.

There are, in fact, some generalizations of the Lagrangian: the Routhian, for example, is a sort of melding between Hamilton and Lagrangian mechanics. And the Law of Least Action, which is more general still, is often used in Quantum Mechanics.

And finally, there is the "master equation." This is a (set) of linear equation(s) that model the time evolution of a system. (I've only been introduced to it in my classes, and I've never sat down to study it on my own, so that's pretty much all I can tell you about this one.)

All of these are related to the Lagrangian, but give a version of Mechanics that is slightly more "robust" than Lagrangian Mechanics. The point to note, though, is that Lagrangian Mechanics (and equivalently Hamiltonian Mechanics) is often general enough to solve any problem we run across. And I have yet to see a problem that can't be solved by the Action Principle, even if only by approximation techniques.

Given that, I (personally) think it is unlikely that anyone is looking for alternatives that scrap the Lagrangian and use a completely new principle.

-Dan

 

[Swamp Thing]

And let's not forget the Kardashian.

 

[vanhees71]

Lagrangian’s are not unique. You can multiply them by a constant or add terms $\frac {dF} {dt}$ for any function F. A magnetic field does no work on a particle, yet you can write down a lagrangian for a particle in a magnetic field.

The function must, however, not depend on the $\dot{q}$! Two Lagrangians $L$ and $L'$ are "equivalent" (i.e., leading to the same equations of motion) if there's a function $F(q,t)$ such that
$$L'=L+\frac{\mathrm{d}}{\mathrm{d}} t F(q,t)=L+ \dot{q} \cdot \frac{\partial F}{\partial q} + \frac{\partial f}{\partial t}.$$
It's easy to see that indeed both Lagrangians lead to the same Euler-Lagrange equations. Start defining the canonical momenta by
$$p'=\frac{\partial L'}{\partial \dot{q}} = \frac{\partial L}{\partial \dot{q}} + \frac{\partial F}{\partial q}=p+\frac{\partial F}{\partial q}.$$
Then the Euler-Lagrange equation for $L'$ read (with the Einstein summation convention)
$$\dot{p}_j'=\dot{p}_j + \dot{q}^k \frac{\partial^2 F}{\partial q^j \partial q^k} + \frac{\partial^2 F}{\partial q^j \partial t}=\frac{\partial L'}{\partial q^j} = \frac{\partial L}{\partial q^j} + \dot{q}_k \frac{\partial^2 F}{\partial q^j \partial q^k} + \frac{\partial^2 F}{\partial q^j \partial t},$$
i.e., the EL equation from $L'$ is just the same as that from $L$,
$$\dot{p}_j = \frac{\partial L}{\partial q^j}.$$

 

[Feynstein100]

Are you asking, "L = T - V is important but how do we know that the "Feynsteinian" ( ) $F = T^2 + 4 V^3 \sqrt{E - 4}$ isn't just as important?"

The answer is that we don't know. Perhaps there is something else out there that we don't yet know about. But that's for the future. If it's out there, I'm sure that someone will (eventually) come across it at some point in time.

There are, in fact, some generalizations of the Lagrangian: the Routhian, for example, is a sort of melding between Hamilton and Lagrangian mechanics. And the Law of Least Action, which is more general still, is often used in Quantum Mechanics.

And finally, there is the "master equation." This is a (set) of linear equation(s) that model the time evolution of a system. (I've only been introduced to it in my classes, and I've never sat down to study it on my own, so that's pretty much all I can tell you about this one.)

All of these are related to the Lagrangian, but give a version of Mechanics that is slightly more "robust" than Lagrangian Mechanics. The point to note, though, is that Lagrangian Mechanics (and equivalently Hamiltonian Mechanics) is often general enough to solve any problem we run across. And I have yet to see a problem that can't be solved by the Action Principle, even if only by approximation techniques.

Given that, I (personally) think it is unlikely that anyone is looking for alternatives that scrap the Lagrangian and use a completely new principle.

-Dan

So it is actually arbitrary and random But that's good. Now I can begin my search for the Feynsteinian

 

[topsquark]

So it is actually arbitrary and random But that's good. Now I can begin my search for the Feynsteinian

It is neither arbitrary, nor random. There is a well defined logic train that leads to the Lagrangian. But that does not mean that there isn't a different train that leads to something different. But, so far as my admittedly limited knowledge base goes, there is no evidence (yet, perhaps) that would lead us in a different direction.

Feel free to look for the Feynsteinian. It might be a fun journey, and you can always learn something from it. And, hey, it might even be out there.

-Dan

 

[Feynstein100]

There is a well defined logic train that leads to the Lagrangian

There is? I'd like to learn more

 

[Vanadium 50]

how do we know that the "Feynsteinian" ( ) isn't just as important?"

Um, because it adds elements of different dimensions?

We know T-V has some utility. We also know T+V has some utility. Is it possible that some other combination has even more utility? Sure (but probably not a linear combination) ... but until someone comes up with one, how will you know for sure?

 

[Frabjous]

There is? I'd like to learn more

You might try looking at
Lanczos
https://www.amazon.com/dp/0486650677/?tag=pfamazon01-20
Rojo and Bloch
https://www.amazon.com/dp/0521869021/?tag=pfamazon01-20
Coopersmith
https://www.amazon.com/dp/0198743041/?tag=pfamazon01-20

 

[topsquark]

There is? I'd like to learn more

See post #4.

-Dan

 

[Feynstein100]

Um, because it adds elements of different dimensions?

Eh......what does that mean?

 

[Feynstein100]

You might try looking at
Lanczos
https://www.amazon.com/dp/0486650677/?tag=pfamazon01-20
Rojo and Bloch
https://www.amazon.com/dp/0521869021/?tag=pfamazon01-20
Coopersmith
https://www.amazon.com/dp/0198743041/?tag=pfamazon01-20

Damn, that's a lot of homework lol. I'll get back to you when I've read them all.