Ad Hoc Lagrangians: Physics & Variational Formulation

[ShayanJ]

Lagrangian mechanics, as you know, is very useful in today's physics. But there is a point that I can't understand.
In cases where we can write L=T-V, Lagrangian mechanics is very useful because for some problems, it gives us a easier way than Newtonian mechanics to derive the equations of motion.
But in cases where we can't write L=T-V, it seems that people just look for a Lagrangian to give the right equations of motion. But that is useless!
For example, we know that for Newtonian gravity we have \nabla^2 \Phi=4\pi G \rho and look for a Lagrangian density to give that same equation and after finding it, they say we have the variational formulation of Newtonian gravity. My question is, why not just use the equation we know? Why should we use something we know to derive something that gives the same thing we knew? That seems non-sense!
Of course its useful for something we're just trying to understand. I mean, when we have no law for some system, we can look for something in it that always gets extremum and then find a variational theory for it and then find the laws of motion.

 

[hilbert2]

As far as I know, the Lagrangian is defined as $L=T-V$. On the other hand, the hamiltonian in not always $H=T+V$, even though it usually is.

 

[ShayanJ]

As far as I know, the Lagrangian is defined as $L=T-V$. On the other hand, the hamiltonian in not always $H=T+V$, even though it usually is.

So you don't know far enough!
Check here!
Another example is the Lagrangian for interaction of a classical charged particle with electromagnetic field.

 

[thegreenlaser]

I don't have an answer to this question, but I just want to mention that I'm interested in it too and I've been waiting for some answers to show up. People say that the Lagrangian approach is somehow "more fundamental," yet the Lagrangian seems to be defined as "something that gives you the right answer." It really seems like there's a piece missing.

Hopefully bringing this back up will stir up some responses...

 

[WannabeNewton]

I don't have an answer to this question, but I just want to mention that I'm interested in it too and I've been waiting for some answers to show up. People say that the Lagrangian approach is somehow "more fundamental," yet the Lagrangian seems to be defined as "something that gives you the right answer." It really seems like there's a piece missing.

If your only exposure to Lagrangian mechanics is through classical mechanics then your viewpoint is fortunately very common amongst students. Indeed at that point it would seem like we just come up with a Lagrangian that gives us the field equations we already knew a priori so the question arises: what's even the point of the Lagrangian, apart from computational facility, given that it is claimed to be more fundamental?

Unless you study more esoteric aspects of classical field theory the answer to this will probably not become apparent on a practical level until you study QFT. The point there is we do not know the field equations beforehand, it has to be actually derived from more fundamental principles of QFT. Indeed we use a very powerful tool called local gauge invariance (or just gauge invariance), along with renormalization, to basically heavily constrain and subsequently guess the Lagrangian for a field theory and from it calculate the field equations.

 

[thegreenlaser]

Unless you study more esoteric aspects of classical field theory the answer to this will probably not become apparent on a practical level until you study QFT. The point there is we do not know the field equations beforehand, it has to be actually derived from more fundamental principles of QFT. Indeed we use a very powerful tool called local gauge invariance (or just gauge invariance), along with renormalization, to basically heavily constrain and subsequently guess the Lagrangian for a field theory and from it calculate the field equations.

Is it possible/useful to do a similar thing in classical mechanics? (I.e., pretend that we don't know the equations already and force ourselves to derive the Lagrangian from more fundamental principles.)

 

[WannabeNewton]

Is it possible/useful to do a similar thing in classical mechanics?

Yes it is possible and often useful.

 

[robphy]

Once you can cast something in a certain form (for example, a Lagrangian),
then you can use the machinery developed to study other aspects that may not be apparent from just the equation of motion (for example: symmetries, generalizations (including dimensionality quantization, etc) , symmetry-preserving computational schemes, recognizing analogies, etc).