- #1

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$$\frac{\partial L}{\partial \dot{q}}=p$$

how to derive that?

On one hand,

$$L=\frac{1}{2}m\dot{q}^2-V$$

so, $$\frac{\partial L}{\partial \dot{q}}=m\dot{q}=p$$

On the other hand,

$$L=\frac{1}{2}m\dot{q}^2-V=\frac{1}{2}m\dot{q}\dot{q}-V=\frac{1}{2}\dot{q}p-V$$

$$\frac{\partial L}{\partial \dot{q}}=\frac{1}{2}p$$

which is half value from the first derivation.