Catenary
Chain under a general force

With no assumptions being made regarding the force G acting on the chain, the following analysis can be made.

First, let T = T (s) be the force of tension as a function of s. The chain is flexible so it can only exert a force parallel to itself. Since tension is defined as the force that the chain exerts on itself, T must be parallel to the chain. In other words,

where T is the magnitude of T and u is the unit tangent vector.

Second, let G = G (s) be the external force per unit length acting on a small segment of a chain as a function of s. The forces acting on the segment of the chain between s and s + Δ s are the force of tension T (s + Δ s) at one end of the segment, the nearly opposite force − T (s) at the other end, and the external force acting on the segment which is approximately G Δ s. These forces must balance so

Divide by Δ s and take the limit as Δ s → 0 to obtain

These equations can be used as the starting point in the analysis of a flexible chain acting under any external force. In the case of the standard catenary, G = (0, − w) where the chain has weight w per unit length.
oulipo:avoid_letter[t], combination:two_responses