How can one say that in Hamilton mechanics the $q$'s are independent of the $p$'s while if I have the Lagrangian $L = \frac{1}{2}\dot{x}^2 + \frac{1}{2}x^2\dot{y}^2$ then $p_y = \frac{\partial{L}}{\partial\dot{y}} = x^2\dot{y}$ and the $p_y$ depends on $x$?
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