# Noether symmetries and conserved quantities for elastic rods with contact

Florence Bertails-Descoubes, Sébastien Neukirch

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Florence Bertails-Descoubes, Sébastien Neukirch. Noether symmetries and conserved quantities for elastic rods with contact. HFSS 2025 - 2nd International Conference on Highly Flexible Slender Structures, Sep 2025, Kaiserslautern, Germany. pp.1-1. hal-05388769

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Submitted on 29 Nov 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Noether symmetries and conserved quantities for elastic rods with contact

Florence Bertails-Descoubes<fnref n="1" />, Sébastien Neukirch<fnref n="2" />

Keywords: structural mechanics, dynamics, integral of motion

## 1. Introduction

The static-dynamic analogy discovered by G. Kirchhoff shows that the statics of an elastic beam are equivalent to the dynamics of a spinning top. In this analogy, time and angular velocity are, for example, equivalent to arc length and curvatures. This static-dynamic analogy allows us to use Noether’s theorem and unravel several quantities that are invariant along the elastic rod at equilibrium. A spinning top having a Lagrangian independent of time will have its mechanical energy constant in time. In the same manner, an elastic rod with uniform elastic properties will have the sum of its curvature energy and its axial force uniform along the structure. The invariant property is known in simple cases, see e.g. [1,2], but the present approach generalizes it to more complex cases where extensibility, shear, gravity, and contact are involved.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Three examples illustrate the use of the invariant in the statics of rods.</figcaption></figure>

We will show how to compute this invariant and illustrate its usefulness in three examples. The first example is the capstan, where an elastic rod is wound around a disk. In this case, the use of the invariant easily leads to the computation of how the opening angle varies with the bending rigidity of the rod and the weight. The second example is the elastic arm scale [3], where the use of the invariant shows that the rod must have a horizontal tangent at the tip where the weight is attached. The third example is the open trefoil knot, where using the invariant leads to the relation between the applied axial force and the loop size.

Finally, we will discuss the generalization of the present approach to the dynamics of 3D twisted rods and its potential for the verification and comparison of numerical simulators for rods.

## Acknowledgments

This project has received funding from CNRS and INRIA.

## References

[1] J. H. Maddocks and D. J. Dichmann. *Conservation laws in the dynamics of rods*. Journal of elasticity, **34** (1994) 83-96

[2] S. Kehrbaum, J. H. Maddocks. *Elastic rods, rigid bodies, quaternions and the last quadrature*. Phil. Trans. Roy. Soc. A, **355** (1997) 2117-2136

[3] F. Bosi, D. Misseroni, F. Dal Corso, and D. Bigoni. *An elastica arm scale*. Proc. Roy. Soc. A, **470** (2014) 20140232.

<footnote n="1">¹ Univ. Grenoble-Alpes, INRIA, CNRS, Grenoble INP, LJK, France, florence descoubes@inria.fr</footnote>

<footnote n="2">² d'Alembert Institute for Mechanics, Sorbonne University, CNRS, Paris, France, neukirch@upmc.fr</footnote>