# Non-holonomic constraints, degree of freedom and generalized coordinates

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?

• if there are K non-holonomic constraints and N degree of freedom, the numbers of the generalized coordinates are N-K – Eli Sep 24 '19 at 8:03
• And degree of freedom? – Harsh Nigam Sep 24 '19 at 16:08
• The degree of freedom are constraint, how many of them is dependent on your physical system – Eli Sep 24 '19 at 19:22

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.
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 $$n\!-\!m/2$$.