Physics GRE - #10

If $\frac{\partial L}{\partial q_n}=0$, where $L$ is the Lagrangian for a conservative system without constraints and $q_n$ is a generalized coordinate, then the generalized momentum $p_n$ is:

• an ignorable coordinate
• constant
• undefined
• equal to $\frac{d}{dt}\left(\frac{\partial L}{\partial q_n}\right)$
• equal to the Hamiltonian for the system

Solution :

The generalized momentum is constant.
Recall that the generalized momentum $p_n$ is \[p_n=\frac{\partial L}{\partial \dot{q_n}}.\]
By the Euler-Lagrange equation, \[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_n}}\right) = \frac{\partial L}{\partial q_n}.\]
Therefore,  \[\frac{\partial L}{\partial q_n}=0\implies\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q_n}}\right) = 0 \implies \frac{\partial L}{\partial\dot{q_n}} = p_n = const.\]