I know that for a lagranian $L=L(q_i, \dot{q_i},t)$ the canonical momentum is given by $p_i = \frac{\partial L}{\partial \dot{q_i}}$. The lagrangian being a function of the generalized coordinate, I would expect $p_i = p_i(q_i, \dot{q_i},t)$. So $q_i$ and $p_i$ are dependent. However according to the Hamiltonian formalism shouldn't $q_i$ and $p_i$ be independent ?
From my research it seems this would be related to the concept of conjugate and canonical variables. These labels are often stated in textbooks with no explications. From which branch of mathematics do these come from ?
This answer refers to manifolds: Independent canonical coordinate variables?
So these concepts come from topology/Differential Geometry/Lie Algebra ?
