We describe an energy-based approach for defining the ``rigidity order'' of a bar and joint framework. We also describe higher order (multivariate) generalizations of the second derivative test from calculus. Together, this gives a new proof of Connelly's theorem, that the lack of a "second order flex" implies rigidity. It gives a new proof of V. Alexandrov's theorem, that when the dimension of non-trivial infinitesimal flexes equals 1, then the lack of a "kth order flex" implies rigidity. When Alexandrov's theorem applies, the rigidity order can be computed just using a sequence of linear algebra computations.