Discussion
If \(p_{j}\) and \(p_{j}'\) are the components of the generalised momenta corresponding to the Lagrangians L and L' respectively, where
\[ L' = L + \frac{d}{dt}F(q_{j}, t), \]
then :
Answer: B
The momenta are related by \(p_{j}' = p_{j} + \frac{\partial F}{\partial q_{j}}\).
Expand the total derivative:
\[ \begin{aligned} \frac{dF}{dt} &= \sum_{k} \frac{\partial F}{\partial q_{k}}\dot{q}_{k} \\ &\quad + \frac{\partial F}{\partial t} \end{aligned} \]
Only the first sum contains the velocities, and it is linear in them, so
\[ \begin{aligned} p_{j}' &= \frac{\partial L'}{\partial \dot{q}_{j}} \\ &= p_{j} + \frac{\partial F}{\partial q_{j}} \end{aligned} \]
Adding a total time derivative leaves the equations of motion unchanged but shifts the canonical momenta, so \(p_{j}' = p_{j}\) is wrong. \(\partial F / \partial t\) has no velocity dependence and drops out, and option D differentiates once too often.
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