This is in reference to the question: Deriving the Lagrangian for a free particle
In Qmechanic'sMy question is specifically in regards to QMechanic's answer to this question, it was claimed that the Lagrangian has to be a differentiable function of velocity and this was used to get rid of one ofI have quoted the branchesrelevant part of the solution.this answer below:
In physics, it is often implicitly assumed that the Lagrangian L=L($\overrightarrow{q}$ ,$\overrightarrow{v}$ ,t) depends smoothly on the (generalized) positions $q_i$, velocities $v_i$, and time t, i.e. that the Lagrangian L is a differentiable function}
However, I am not sure whyWhy should we demand thatassume the Lagrangian has to be a differentiable with respect to velocities. In fact, the first linefunction of this answer also claims that the Lagrangian must be differentiable in the coordinates, which isn't always true.positions and velocities?
For example, every potential of the form $1/r^n$ is non-differentiable at the origin, for $n>0$ and we do encounter such potentials all the time (for $n=1$, we recover the usual central force problem).
Sorry if this has been asked before or ifCan we not have any situations where the correct procedure was to ask itLagrangian is not a differentiable function of the velocities?
This is relevant because later in the comments underneath that answer, they say:
This is differentiable wrt. the speed v=$|\overrightarrow{v}|$, but it is not differentiable wrt. the velocity $\overrightarrow{v}$ at $\overrightarrow{v}=\overrightarrow{0}$ if $\alpha \neq 0$. Therefore the second branch (6) is discarded.
That is, they are using the non-differentiability of a candidate Lagrangian with respect to velocities to rule it out.
