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From Lifshitz and Landau Vol.$1$:

From the equation in differentials $$ \mathrm{d} H=-\sum \dot{p}_{i} \mathrm{d} q_{i}+\sum \dot{q}_{i} \mathrm{d} p_{i} $$ in which the independent variables are the co-ordinates and momenta, we have the equations $$ \dot{q}_{i}=\partial H / \partial p_{i}, \quad \dot{p}_{i}=-\partial H / \partial q_{i} $$

The problem is that I do not know how to work with differentials in this situation. After messing around I get that the right hand side just cancels out and $\partial H / \partial p_{i}=0.$ Any suggestions appreciated.

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If $H(q,p)$ is any function at all, then its differential is

$$dH = \frac{\partial H}{\partial q} dq + \frac{\partial H}{\partial p} dp,$$

simply by definition of what a differential is. If we also know from elsewhere that

$$dH = -\dot{p}\, dq + \dot{q}\, dp,$$

then just by comparing the coefficients of the differentials we can see that

$$- \dot{p} = \frac{\partial H}{\partial q} \quad \text{and} \quad \dot{q} = \frac{\partial H}{\partial p}.$$

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