Suppose we consider a rigid body, which has $N$ particles. Then the number of degrees of freedom is $3N – (\mbox{# of constraints})$.
As the distance between any two points in a rigid body is fixed, we have $N\choose{2}$ constraints giving $$\mbox{d.o.f} = 3N – \frac{N(N-1)}{2}.$$ But as $N$ becomes large the second term being quadratic would dominate giving a negative number. How do we explain this negative degrees of freedom paradox?
Best Answer
You've duplicated constraints because if any one particle is constrainined in all three dimensions with all the other particles this constrains all the particles. The number of constraints is 3(N - 1).
To give an example, take three particles a, b and c. If a is fixed relative to b and is also fixed relative to c, then b and c are fixed relative to each other without having to introduce new constraints.