Finding Lagrangian: General Procedure?

[ShayanJ]

In texts about Lagrangian mechanics,at first Lagrangian is defined as below:
$L=T-V$
T and V being kinetic and potential energy respectively
But when you proceed,they say that for some forces like magnetic forces Lagrangian is as such and can't be obtained by the above formula but it doesn't say how it obtains the Lagrangian for a particle subjected to magnetic forces.
I want to know is there a general procedure for finding the Lagrangian?
Thanks

 

[TobyC]

In texts about Lagrangian mechanics,at first Lagrangian is defined as below:
$L=T-V$
T and V being kinetic and potential energy respectively
But when you proceed,they say that for some forces like magnetic forces Lagrangian is as such and can't be obtained by the above formula but it doesn't say how it obtains the Lagrangian for a particle subjected to magnetic forces.
I want to know is there a general procedure for finding the Lagrangian?
Thanks

The Lagrangian for the electromagnetic force is I think just a fundamental law of physics, you can't derive it from anything more fundamental. Just like in the Newtonian way of doing mechanics you take Newton's law of gravity or the Lorentz force on a charged particle as fundamental, you take the Lagrangian for these forces as fundamental in Lagrangian mechanics. The L = T - V formula just gives you a quick way of going from forces that are described by a simple potential (V) in Newtonian mechanics to Lagrangian mechanics. The electromagnetic force is not describable by a single potential field so that formula doesn't work.

However, although you can't really derive the electromagnetic lagrangian from anything more fundamental, you can show that it is the lagrangian you need in order to reproduce the same results that you get in the Newtonian description of electromagnetism. Starting with the Lorentz force expression, you can derive the electromagnetic lagrangian, which is sort of deriving it from something equally fundamental rather than more fundamental. To do that just fiddle with the lorentz force expression by first writing it in terms of the scalar and vector potentials rather than E and B fields and then try and arrange it into a form that is the same as Lagrange's equations of motion and look at what the Lagrangian is.

 

[ShayanJ]

Thanks
But what about other situations where we can't use L=T-V?(Are there any?)

 

[TobyC]

Thanks
But what about other situations where we can't use L=T-V?(Are there any?)

Well it would just be any situation where the force isn't just simply the gradient of some potential. I can't think of a simple example other than electromagnetism. But actually any force that can be described with a single potential will not be compatible with relativity, so once relativity becomes important no Lagrangians will be of the form T - V. In Quantum Field Theory for example I expect that formula becomes pretty redundant, but I don't know any Quantum Field Theory. In classical mechanics I think the T - V will basically always work unless you're dealing with electromagnetism, I'm not an expert though.

 

[PJC01]

for your question. The general procedure for finding the Lagrangian in Lagrangian mechanics involves considering the kinetic and potential energy of a system. However, as you mentioned, there are some cases where the Lagrangian cannot be obtained using this formula alone, such as when there are magnetic forces involved.

In these cases, the Lagrangian can be obtained by considering the work done by the magnetic forces on the system. This work can be expressed in terms of the magnetic vector potential, which is a fundamental quantity in electromagnetism.

In general, the Lagrangian can be found by considering all the forces acting on a system and expressing them in terms of generalized coordinates and their derivatives. This approach allows for a more comprehensive understanding of the dynamics of a system and can be applied to a wide range of physical systems.

It is important to note that finding the Lagrangian is not always a straightforward process and may require specialized knowledge and techniques in certain cases. However, the general procedure remains the same – consider all the forces acting on the system and express them in terms of generalized coordinates to obtain the Lagrangian.

I hope this helps clarify the general procedure for finding the Lagrangian in Lagrangian mechanics. Please feel free to ask any further questions.