Talk:Relativistic Lagrangian mechanics

Clarifying energy vs. coenergy
Strictly speaking, the Lagrangian for a relativistic system is defined as L = T* - V, where T* is the kinetic coenergy function. The kinetic coenergy function is equal to kinetic energy for classical mechanics, but differs in relativistic systems. The article on Lagrangian mechanics does not clarify this distinction, so I believe it should be clarified here.

Transfer of content from Lagrangian mechanics
The new content was originally in the Lagrangian mechanics article, most of which has been rewritten by me at the time of this note. Since that article became too long, I moved it here. This article was a hideous redirect to... an article in string theory. The new content is fitting for the title. Most of the examples were originally written by others for the older version of the Lagrangian article (now a disambiguation page), which were transferred by me into Lagrangian mechanics, now they are here. M&and;Ŝc2ħεИτlk 12:42, 21 September 2015 (UTC)

Correction?
Is this equation correct?


 * $$ L(\tau) = \frac{1}{2}m u^\mu(\tau)u_\mu(\tau) + qu^\mu(\tau)A_\mu(x)$$

As far as I can tell, it doesn't equal the Lagrangian given above:


 * $$ L = - m c^2 \sqrt {1 - \frac{v^2 }{c^2}} - q \phi + q \dot{\mathbf{r}} \cdot \mathbf{A} \,.$$

The former equation implies another factor of $$\gamma$$ on the latter, right? And the factor of $$\frac{1}{2}$$ seems like it should be there either. There might also be a missing sign but I'm not sure what metric signature is being used as it's not stated.

I'm hesitant to edit it without finding a source but wanted to ask. Shevvvv (talk) 23:35, 6 September 2023 (UTC)


 * No. Your first equation is wrong, indeed ridiculous. You must have a reliable secondary source for any equation you add to the article. JRSpriggs (talk) 04:22, 8 September 2023 (UTC)
 * There's no reason to be rude.
 * My point is that the first equation is in the article in the section "Special relativistic test particle in an electromagnetic field" and seems completely wrong. I wanted to correct it but don't have a reference available for the corrected form. Shevvvv (talk) 20:52, 6 February 2024 (UTC)

Can d'Alembert's principle derivation of relativistic Lagrangian replace the convoluted derivation in this article?
Which goes something like this. EditingPencil (talk) 20:33, 24 November 2023 (UTC)


 * I have a related question: why do we have this lengthy unreferenced "derivation" at all?
 * The article should give the essence of relativistic lagrangian mechanics, not a textbook regurgitation. That's what references or Further Reading is for. Johnjbarton (talk) 00:30, 21 February 2024 (UTC)
 * I agree, it doesn't matter what type of derivation that the form of the Lagrangian naturally arises in. I edited justification to strengthen the same argument here. In hindsight, this was a hastily written question, lol. EditingPencil (talk) 08:01, 21 February 2024 (UTC)
 * Link to diff of my edit. EditingPencil (talk) 08:03, 21 February 2024 (UTC)