How to derive the linear form of Pfaffian equation?

Question

I am working on fundamental dynamics in last few days. The question is about Pfaffian constraint.

A general form of Pfaffian constraint is $$ A(q)^{\mathrm{T}}d{q} + b(q)dt = 0\tag{1}$$ which is formed as $$ A(q)^{\mathrm{T}} \delta q = 0\tag{2}$$ with variation theorem, i.e. $$b(q)dt=0.\tag{3}$$

However, in some applications such as non-holonomic system, that the Pfaffain form is simply given by $$A(q)^{\mathrm{T}}\dot{q} = 0.\tag{4}$$ I simply want the detailed derivation of this form. After searching internet, I didn't find too much things helpful to the derivation of this form.

Answer 1

1. The semi-holonomic/Pfaffian constraint (1) is equivalent to $$ A(q)^{\mathrm{T}}\dot{q} +b(q)~=~ 0;\tag{1'} $$ not eq. (4).

2. However, when considering d'Alembert's principle, time is frozen $$\delta t~=~0,$$ so that an infinitesimal virtual displacement $\delta q$ satisfies eq. (2).