Higher Order Rigidity and Energy

TL;DR

This paper explores the relationship between higher-order rigidity of frameworks and energy functions, introducing a new general definition of rigidity order and applying higher derivative tests to establish rigidity conditions.

Contribution

It introduces a novel, energy-independent definition of rigidity order and demonstrates its application using higher order derivative tests, including new proofs of rigidity criteria.

Findings

Lack of second order flex implies rigidity

Frameworks with 1D first-order flexes and no higher order flex are rigid

Higher derivative tests can be applied to analyze rigidity

Abstract

In this paper, we revisit the notion of higher-order rigidity of a bar-and-joint framework. In particular, we provide a link between the rigidity properties of a framework, and the growth order of an energy function defined on that framework. Using our approach, we propose a general definition for the rigidity order of a framework, and we show that this definition does not depend on the details of the chosen energy function. Then we show how this order can be studied using higher order derivative tests. Doing so, we obtain a new proof that the lack of a second order flex implies rigidity. Our proof relies on our construction of a fourth derivative test, which may be applied to a critical point when the second derivative test fails. We also obtain a new proof that when the dimension of non-trivial first-order flexes equals $1$, then the lack of a $k$th order flex for some $k$ implies a framework is rigid. The higher order derivative tests that we study here may have applications beyond rigidity theory.