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Thursday, 11 August 2011

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,  \[\frac{\partial L}{\partial q_n}=0\implies\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q_n}}\right) = 0 \implies \frac{\partial L}{\partial\dot{q_n}} = p_n = const.\]


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