Problem/question

Using definition of Levi-Civita  epsilon symbol and Kronecker delta symbol to show that \begin{equation}\epsilon_{i\,j\,k}\,\epsilon_{k\,\ell\,m}\ =\ (\,\delta_{i\ell}\,\delta_{jm}\,-\,\delta_{im}\,\delta_{j\ell}) \end{equation} Use this  identity to prove that \begin{equation}\vec{A}\times(\vec{B}\times\vec{C})=(\vec{A}\cdot\vec{C})\vec{B} -(\vec { A } \cdot\vec{B})\vec{C}.\end{equation}

1. Consider a system of three point masses connected by rigid rods. Find the number of constraints and hence the number of generalized coordinates required for this system when the masses do not lie in a straight line.
2. Next repeat the above problem for three three masses constrained to lie in a straight line.
3. Count the number of constraints and hence find the number of degrees of a freedom of a rigid body consisting of \(n\) point masses separated by fixed distances. Compute the value of number of degrees of freedom for a few values of \(n\) . What happens when \(n\) is very large?

A useful reference for counting of degrees of freedom for a rigid body, in a manner described above, is
Jorge Bernal et al., "Exact calculation of the number of degrees of freedom of a rigid body constituted by n particles".  arXiv:1002.2002v1 (Feb9,2010)

Question