The total time derivative of a differentiable function $g(q_i,t)$ is added to a Lagrangian: $$L\prime=L(q_i,\dot q_i,t)+\frac{d}{dt}g(q_i,t).$$ Let $p_i,H$ and $p_i\prime,H\prime$ be the respective generalized momenta and Hamiltonians. Which statements are correct?
$L$ and $L\prime$ give the same Euler–Lagrange equations of motion.
$p_i\prime=p_i+\partial g/\partial q_i$
If $p_i$ is conserved, $p_i\prime$ is necessarily conserved.
Correct answer: (A) $L$ and $L\prime$ give the same Euler–Lagrange equations of motion.; (B) $p_i\prime=p_i+\partial g/\partial q_i$
Explanation
Expand $dg/dt=\sum_i(\partial g/\partial q_i)\dot q_i+\partial g/\partial t$. Its contribution to the action is an endpoint term, so fixed-endpoint variations give the same equations of motion: A. Differentiating with respect to $\dot q_i$ gives $p_i\prime=p_i+\partial g/\partial q_i$: B. This additional derivative can vary in time, so C is not necessary. The Legendre transform gives $H\prime=H-\partial g/\partial t$, evaluated at corresponding momenta, not D.
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