The Hamiltonian is defined as the Legendre transform of the Lagrangian $$H = p\dot{q} -L .$$ In the Lagrangian formalism we are free to add the total derivative of an arbitrary function $F=F(q,t)$ to the Lagrangian $$ L \to L' = L + \frac{dF}{dt} $$ because such an additional term has no effect on the Euler-Lagrange equation.   For the momentum $ p =  \frac{\partial L}{\partial \dot{q}} $ such a shift of the Lagrangian implies $$ p \to p'= p +  \frac{\partial  F}{\partial q} .$$   If we combine these two observations, we find the following transformation law for the Hamiltonian \begin{align} H(q,p) \to H' &amp;= p' \dot q' - L' \\ &amp;= \left( p + \frac{\partial  F}{\partial q} \right) \dot q - \Big( L + \frac{dF}{dt} \Big) \\ &amp;= H + \frac{\partial  F}{\partial q} \dot q - \frac{dF}{dt}  \end{align} Is this transformation law correct or am I missing something important? (In the literature I've only found several times the statement that the Hamiltonian like the Lagrangian is allowed at most to change be the total derivative of some function. But here it seem that there is an additional term...)
The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's Classical Mechanics.   Adding the total time derivative of a function of $q_i$ and t to the Lagrangian will not change the the Euler-Lagrangian equation. So if we make the following change to Lagrangian,   $$L'(q,\dot{q},t) = L(q,\dot{q},t) + \frac{dF(q_1,q_2,...,q_n,t)}{dt}$$ we can get  $$ \frac{d}{dt}\frac{\partial{L'}}{\partial{\dot{q_i}}} - \frac{\partial{L'}}{\partial{q_i}} = 0 $$ from  $$ \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q_i}}} - \frac{\partial{L}}{\partial{q_i}} = 0 $$  How can we get the corresponding Hamiltonian equation part? This is to prove  $$ \dot{p'_i} = \frac{\partial{H'}}{\partial q_i} $$ $$ -\dot{q_i} = \frac{\partial{H'}}{\partial p'_i}  $$  from   $$ \dot{p_i} = \frac{\partial{H}}{\partial q_i} $$ $$ -\dot{q_i} = \frac{\partial{H}}{\partial p_i}  $$ where $p'_i = \frac{\partial L'}{\partial \dot q_i}$.  Edit  The corresponding $H'$ is $$ H' = \sum_k{p'_k \dot{q_k}} - L' $$ where $p'_k = \frac{\partial L'}{\partial \dot q_k}$.
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