# Simple lagrangian question

iScience
hi, silly question but would someone please show me how $\frac{∂L}{∂q}$=$\dot{p}$?

L being the lagrangian, p being the momentum, and q being the general coordinate.

iScience
(i hope this doesn't qualify as a homework question this actually has nothing to do with my homework.. i'm just trying to derive the hamiltonian equation and this was just part of the steps)

Gold Member
please show me how $\frac{∂L}{∂q}$=$\dot{p}$?

$\frac{∂L}{∂q}$=$\dot{p}$ because the Lagrangian, L=T-U, depends on q only in the potential energy, U.

$-\frac{∂U}{∂q}$=$\dot{p}$ is Newton's 2nd law of motion expressed in terms of the potential.

iScience
thanks!!

also, i had another question i hope i can just ask it in the same thread; if not let me know (moderators/admins) and i'll just make a new thread.

how do i know when there is translational symmetry? in other words what quantity has to be zero?

Gold Member
You want p=constant (p-dot = 0) for the momentum conjugate to that coordinate.

iScience
"momentum that is conjugate to that coordinate" meaning just the momentum corresponding to the particular coordinate at hand right? (just checking)

The canonical momentum of the generalized coordinate $q$ is by definition given by
$$p=\frac{\partial L}{\partial \dot{q}}.$$