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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?

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  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/8903 it might answer your questions $\endgroup$ – user139561 Dec 18 '16 at 22:55
  • $\begingroup$ Note that to get the constraint forces you must apply the Method of Undetermined Multipliers, but at least that is a algorithmic. $\endgroup$ – dmckee Dec 19 '16 at 0:08
  • $\begingroup$ Examples of constraint forces that you know and use in a Newtonian context are normal forces and static friction. They each impose a constraint ('two bodies can't occupy the same space' and 'no relative motion' respectively) and you have to put them into your consideration as unknowns. In sufficiently simple case you can find them, but in more complex situations this can be hard. $\endgroup$ – dmckee Dec 19 '16 at 4:50
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My specific answer is as follows:

a) If (2) is resolved, (1) would not strictly be a problem, but still a big inconvenience, as you would have to solve more number of equations with more variables, compared to generalized coordinates.

For example, if you know that the boy moves in a circle, your generalized coordinate is $q=\theta$, as the radius $R$ is constant, but using Newton's equations, you have to deal with the $x$ and $y$ coordinate, along with the equation of constraint $x^2+y^2=R^2$ (assuming center of circle as origin).

b) Newton's law says that $F=\dfrac{dp}{dt}=ma$ (obviously non-relativistic)

The constraint can be taken care of either by an extra condition like above circular case, or by specifying the constraint force which keeps the body in it's path. As the force is almost always unknown, using the constraint condition is necessary. As you said, the Newton's law equation would not be correct. Actually the law is always correct, but if forces or conditions are missing, it won't give correct answers, so the application of the law is wrong in such cases.

A trivial point: If the constraint force is zero, for example, the force is $m\dfrac{v^2}{r}$, then no extra conditions are necessary, as the body is moving in circle not because of any special constraint, but purely due to the force. The same situation can be solved in terms of constraints too, but I won't go into that.

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