Homology, equilibrium, and conservation laws I: Discrete systems of points

TL;DR

This paper reviews how homology and cohomology methods can be applied to analyze electrical and mechanical networks, revealing the topological and conservation law foundations of circuit and system equilibrium.

Contribution

It introduces a homological perspective to electrical and mechanical network analysis, connecting conservation laws with topological structures.

Findings

Kirchhoff's laws are inherently homological.

Homological methods clarify conservation laws in networks.

Topological analysis applies to both electrical and mechanical systems.

Abstract

The methods of abstract simplicial homology and cohomology are reviewed and applied to the topology of electrical networks. Kirchhoffs laws of electrical circuits are shown to be manifestly homological in their origins. Since they are based in conservation laws, the geometric realization of abstract simplicial complexes is then reviewed and applied to the case of mechanical networks. The equilibrium condition for statics, the conservation laws for closed systems, and the balance principles for open systems are then shown to admit homological formulations.