Why do some of the terms of the gradient of the dot product between the velocity and magnetic vector potential disappear? When using the Euler-Lagrange equations to yield the Lorentz force, one of the terms ends up looking like this 
$$∂L/∂r =q(∇(v⋅A)−∇ϕ) =q(v×(∇×A)+(v⋅∇)A−∇ϕ)$$
Where $v$ is the velocity and $A$ is the magnetic vector potential
The gradient of dot product equals,
$$∇(a⋅b)=(a⋅∇)b+(b⋅∇)a+a×(∇×b)+b×(∇×a)$$
So the equation would be missing the terms in bold $$∂L/∂r=q(∇(v⋅A)−∇ϕ)=q(v×(∇×A)+(v⋅∇)A+(\mathbf{A⋅}\mathbf{∇})\mathbf{v}+\mathbf{A}\mathbf{×}(\mathbf{∇×}\mathbf{v})−∇ϕ)$$
What's the reason that these absent terms disappear?
 A: You are ignoring that you are working in the $2N+1$ dimensional phase space $\{\boldsymbol q, \boldsymbol {\dot q}, t\}$. It's only along the locus of points in this phase space that satisfy the Euler-Lagrange equations that $\boldsymbol {\dot q}$ is indeed the time derivative of $\boldsymbol q$. Until you arrive at that solution, you must treat $\frac{\partial \dot q_j}{\partial q_i}$ and $\frac{\partial q_j}{\partial \dot q_i}$ as being identically equal to zero. This means that $$\nabla (\boldsymbol A \cdot \boldsymbol v) =  \sum_i \hat e_i \frac{\partial}{\partial r_i} \left(\sum_j A_j  \dot r_j\right) = \sum_i \hat e_i \sum_j \frac{\partial A_j}{\partial r_i} \dot r_j \equiv \hat e_i \frac{\partial A_j}{\partial r_i} \dot r_j$$
where the double summation is implied in the rightmost term. Note well: There are no $A_j \frac{\partial \dot r_j}{\partial r_i}$ terms.

Another way to look at this: Using $\partial/\partial q_i$ and $\partial/\partial \dot q_i$ is an abuse of notation in this context (and even worse, using $\nabla$) because the generalized velocity generally varies with generalized position along the solution to the Euler-Lagrange equations. In other words, $\nabla v$ is not identically zero. As with other mathematical abuses of notation, this notation can be very handy, but it will get you in trouble if you don't realize that it is an abuse of notation.
