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This paper explores the equilibria of small linear translational spring networks using advanced techniques from nonlinear algebra, specifically homotopy continuation, monodromy, and parameter homotopy. The authors implement a novel parameter homotopy method that leverages the physical structure of the spring systems, achieving significant computational efficiency compared to traditional methods like Newton's method. Key findings include precise total degree bounds on the maximum number of solutions for planar spring networks and enhanced efficiency through polyhedral homotopy methods, demonstrating the potential for improved analysis of mechanical systems.
Rigid spring networks can yield a surprising number of equilibrium solutions, with new methods cutting computation time significantly compared to traditional approaches.
We use tools from nonlinear algebra to study the equilibria of small linear translational spring networks. Specifically we use the techniques of homotopy continuation, monodromy, and parameter homotopy (a.k.a. cheater homotopy) to solve all rigid linear translational spring networks up to $5$ nodes in both $2$ and $3$ dimensions. We describe a method of implementing parameter homotopy that arises naturally from the physical structure of the system. We give precise total degree bounds on the maximum number of solutions for general planar spring networks. We discuss further efficiency gains obtained from polyhedral homotopy methods. We compare the computation efficiency of these techniques against a baseline of Newton's method.