It's told in Landau – Classical Mechanics, that in the Hamiltonian method, generalized coordinates $q_j$ and generalized momenta $p_j$ are independent variables of a mechanical system. Anyway, in the case of Lagrangian method only generalized coordinates $q_j$ are independent. In this case generalized velocities are not independent, as they are the derivatives of coordinates.
So, as I understood, in the first method, there are twice more independent variables, than in the second. This fact is used during the variation of action and finding the equations of motion.
My question is, can the number of independent variables of the same system be different in these cases? Besides that, how can the momenta be independent from coordinates, if we have this equation $$p=\frac{\partial L}{\partial \dot{q}}.$$
Best Answer
1L) The (generalized) position $q$ and (generalized) velocity $v$ are independent variables of the Lagrangian $L(q,v,t)$, cf. e.g. this Phys.SE post.
1H) The position $q$ and momentum $p$ are independent variables of the Hamiltonian $H(q,p,t)$.
2L) The position path $q:[t_i,t_f] \to \mathbb{R}$ and velocity path $\dot{q}:[t_i,t_f] \to \mathbb{R}$ are not independent in the Lagrangian action $$S_L[q]~=~ \int_{t_i}^{t_f}\!dt \ L(q ,\dot{q},t).$$ See also this Phys.SE post.
2H) The position path $q:[t_i,t_f] \to \mathbb{R}$ and momentum path $p:[t_i,t_f] \to \mathbb{R}$ are independent in the Hamiltonian action $$S_H[q,p]~=~\int_{t_i}^{t_f}\! dt~(p \dot{q}-H(q,p,t)).$$
3L) Under extremization of the Lagrangian action $S_L[q]$ wrt. the path $q$, the corresponding equation for the extremal path is Lagrange's equation of motion $$\frac{d}{dt}\frac{\partial L(q,\dot{q},t)}{\partial \dot{q}} ~=~ \frac{\partial L(q,\dot{q},t)}{\partial q},$$ which is a second-order ODE.
3H) Under extremization of the Hamiltonian action $S_H[q,p]$ wrt. the paths $q$ and $p$, the corresponding equations for the extremal paths are Hamilton's equations of motion $$-\dot{p}~=~\frac{\partial H}{\partial q} \qquad \text{and}\qquad \dot{q}~=~\frac{\partial H}{\partial p} ,$$ which are first-order ODEs.
4L) The equation $p=\frac{\partial L}{\partial v}$ is a definition in the Lagrangian formalism. E.g., for a non-relativistic free point particle, it encodes the relation $p=mv$.
4H) The equation $\dot{q}=\frac{\partial H}{\partial p}$ is an equation of motion in the Hamiltonian formalism. E.g., for a non-relativistic free point particle, it encodes the relation $p=m\dot{q}$.