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## Homework Statement

This is derivation 2 from chapter 8 of Goldstein:

It has been previously noted that the total time derivative of a function of ## q_i## and ## t ## can be added to the Lagrangian without changing the equations of motion. What does such an addition do to the canonical momenta and the Hamiltonian? Show that the equations of motion in terms of the new Hamiltonian reduce to the original Hamilton's equations of motion.

## Homework Equations

Euler-Lagrange equations:

$$ \frac{\mathrm{d}}{\mathrm{d} {t}} \Big(\frac{\partial L}{\partial \dot{q}_i}\Big) = \frac{\partial L}{\partial q_i}$$

Hamilton's equations:

$$ \frac{\partial H}{\partial p_i} = \dot{q}_i$$

$$\frac{\partial H}{\partial q_i} = -\dot{p}_i$$

## The Attempt at a Solution

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Let $$L' = L + \frac{\mathrm{d} F}{\mathrm{d} t} $$ where ## F = F(\vec{q}, t)##

$$ L' = L + \frac{\partial F}{\partial q_k} \dot{q}_k + \frac{\partial F}{\partial t}$$

$$p_i' = \frac{\partial L'}{\partial \dot{q}_i} = p_i + \frac{\partial F}{\partial q_i}$$

$$ H' = p'_i \dot{q}_i - L' = p_i \dot{q}_i - L - \frac{\partial F}{\partial t} = H - \frac{\partial F}{\partial t} $$

$$ \frac{\partial H'}{\partial p_j '} = \frac{\partial H'}{\partial p_k}\frac{\partial p_k}{\partial p_j'} = \frac{\partial H}{\partial p_j} = \dot{q}_j$$

$$\frac{\partial H'}{\partial q_j} = \frac{\partial H}{\partial q_j} - \frac{\partial ^2 F}{\partial q_j \partial t} = -\dot{p}_j - \frac{\partial ^2 F}{\partial q_j \partial t}$$

But

$$\dot{p}'_j = \dot{p}_j + \frac{\mathrm{d}}{\mathrm{d} t} \Big( \frac{\partial F}{\partial q_j} \Big) = \dot{p}_j + \frac{\partial ^2 F}{\partial q_j \partial t} + \frac{\partial ^2 F}{\partial q_i \partial q_k} \dot{q}_k$$

so $$ \frac{\partial H'}{\partial q_j} \neq -\dot{p}'_j $$

I feel like I am making a simple mistake here but I cannot spot it. Any help would be much appreciated.

Thank you in advance!

Barek