Non-holonomic constraints, degree of freedom and generalized coordinates

Question

If a system has $N$ coordinates and $M$ number of holonomic constraints then number of degree of freedom $=N-M$ and generalized coordinates $=N-M$ too. But if there are $k$ non-holonomic constraints then what will be no. of degree of freedom and generalized coordinates?

Answer 1

1. Consider a classical point-mechanical system with $3N$ coordinates but only $n$ generalized coordinates $(q^1, \ldots,q^n)$, because of $3N-n$ holonomic constraints.

2. Let us for simplicity assume that:

   • All constraints are 2-sided, i.e. we do not allow 1-sided constraints (= inequalities).

   • All non-holonomic constraints are semi-holonomic.

3. If furthermore the system has $m$ semi-holonomic constraints, then we introduce $m$ Lagrange multipliers $(\lambda^1, \ldots,\lambda^m)$. The corresponding $n$ Lagrange equations are outlined in this Phys.SE post.

4. The number of degrees of freedom (DOF) are conventionally defined as half the number of initial conditions needed for $(q^1, \ldots,q^n,\dot{q}^1, \ldots,\dot{q}^n)$, which is then $n\!-\!m/2$, which curiously could be a half-integer. (The main point is that since each semi-holonomic constraint is a 1st-order ODE, they each remove the need of 1 initial condition.)