A conserved quantity in thin body dynamics

•A conserved quantity relevant to the dynamical equilibria of thin structures.•A mixed Lagrangian–Eulerian non-material action principle for fixed windows of axially moving systems.•Analytical solutions for rotating, flowing strings (yarn balloons).•Noether meets Bernoulli in a textile factory.

Thin, solid bodies with metric symmetries admit a restricted form of reparameterization invariance. Their dynamical equilibria include motions with both rigid and flowing aspects. On such configurations, a quantity is conserved along the intrinsic coordinate corresponding to the symmetry. As an example of its utility, this conserved quantity is combined with linear and angular momentum currents to construct solutions for the equilibria of a rotating, flowing string, for which it is akin to Bernoulli's constant.