I am reading the book "Classical Mechanics" by Douglas Gregory, and the author writes that using Newtonian equations for constrained systems runs into two difficuties.
(1). The equations of motion do not incorporate the constraints The Newton equations (in Cartesian coordinates) do not incorporate the constraints. These must therefore be included in the form of additional conditions to be solved simultaneously with the dynamical equations.
(2). The constraint forces are unknow
And he also writes
(1) is overcome by generalized coordinates while (2) is overcome by using Lagrange's equations instead of Newton's
My questions are:
(a) Isn't (1) and (2) essentially the same difficulty? I mean, (1) wouldn't be a problem if (2) is resolved, would it? Why does the author distinguish (1) and (2)?
(b) I am confused about what is said in (1). If we write the Newton equations without incorporating the constraints, the equations are "wrong" in essence. Aren't they? As far as I know, the Newton equations are true only when all the forces are identified. So, is the author saying that the "wrong" newtons equations with constraint equations are equivalent to "correct" newton's equations?