ESPCI web site
Elastic Rods made easy with a Noether invariant
The static-dynamic analogy discovered by G. Kirchhoff shows that the statics of the planar elastica are equivalent to the dynamics of the pendulum. In this analogy, time and angular velocity are, for example, equivalent to arc length and curvatures.
This static-dynamic analogy allows us to write a quantity that is invariant along the elastic rod at equilibrium : A pendulum will have its mechanical energy constant in time and, in the same manner, a planar elastica will have the sum of its curvature energy and its axial force uniform along the structure.
We will show how this invariant may be used to quickly derive useful information on the equilibrium shapes of elastica in self-contact or in interaction with obstacles, sliding sleeves, force fields, and droplets.