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Equilibria for Networks of Linear Translational Springs
Abstract
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.
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BibTeX
@misc{oeding2026equilibria,
title = {{Equilibria for Networks of Linear Translational Springs}},
author = {Luke Oeding and Ethan Clayton and Jackson Elsea and Nicholas Wang and Adam Rutkowski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03143},
eprint = {2609.03143},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03143},
}
RIS
TY - UNPB
TI - Equilibria for Networks of Linear Translational Springs
AU - Luke Oeding
AU - Ethan Clayton
AU - Jackson Elsea
AU - Nicholas Wang
AU - Adam Rutkowski
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.03143
UR - https://arxiv.org/abs/2609.03143
AB - 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.
ER -