Is there a more fundamental reason for the Classical Lagrangian to be T−V and Electromagnetic Lagrangian to be T−V+qA.v or is it simply because we can derive Newton's Second Law and Lorentz Force Law with these Lagrangians respectively?

Jaxon Hamilton 2022-07-20 Answered
Is there a more fundamental reason for the Classical Lagrangian to be T−V and Electromagnetic Lagrangian to be T−V+qA.v or is it simply because we can derive Newton's Second Law and Lorentz Force Law with these Lagrangians respectively?
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Wayne Everett
By defining the lagrangian as
you're restricting yourself. This definition works only in classical mechanics.
A lagrangian is a vastly more general concept which I'll summatrize: $L\left(q\left(t\right),\eta \left(t\right),t\right)$ is an arbitrary differentiable map
$L:A×\mathbb{R}\to \mathbb{R}\phantom{\rule{2em}{0ex}}A\subseteq {\mathbb{R}}^{n}×{\mathbb{R}}^{n}$
where $q\left(t\right)$ are generalized coordinates and $\eta \left(t\right)$ are generalized velocities. With this we define the action functional
$S\left[q,\eta \right]={\int }_{{t}_{0}}^{t}L\left(q\left(t\right),\eta \left(t\right),t\right)dt$
Then, basically all physics can be derived from Hamilton principle of stationary action which says that
By evaluating the variation of the action $\delta S$ and by imposing the stationary condition one finds that a path q(t) makes the action stationary if is a solution to the equation
$\frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }{\eta }_{i}}\left(q\left(t\right),\stackrel{˙}{q}\left(t\right)\right)-\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_{i}}\left(q\left(t\right),\eta \left(t\right)\right)=0\phantom{\rule{0ex}{0ex}}det\left(\frac{{\mathrm{\partial }}^{2}L}{\mathrm{\partial }{\eta }_{i}\mathrm{\partial }{\eta }_{j}}\right)\ne 0$
One can then use the change of variables $\stackrel{˙}{q}\left(t\right)=\eta \left(t\right)$. Note that in all of this we never talked about a specific form of the lagrangian such as T-V
This general formulation of a lagrangian is what is used everywhere in physics and can be extended even to quantum theories like quantum field theory. Even though in QFT we mostly speak of lagrangian densities, which does not change much.