Sébastien Neukirch
Institut Jean le Rond d'Alembert
Centre National de la Recherche Scientifique
Sorbonne Université, Campus Pierre et Marie Curie
Paris, Francetel: +33 1 44 27 72 61
e-mail: sebastien.neukirch (-atat-) upmc.fr
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Sébastien Neukirch and Florence Bertails-Descoubes
A Noetherian approach to invariants for the statics and dynamics of elastic rods
---, --- (2025) pp---
Abstract : The static-dynamic analogy discovered by G. Kirchhoff shows that the statics of an elastic beam is equivalent to the dynamics of a spinning top. This analogy, where time and angular velocity are, for example, equivalent to arclength and curvatures, allows the use of Noether's theorem to unravel a quantity that is invariant along elastic rods 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 tension force uniform along the structure. The invariant property is known in simple cases, but the present approach generalises it to more complex cases where extensibility, shear, conservative loads (e.g. gravity), and contact are involved. Moreover, still using Noether's theorem and bringing to light the continuous symmetries of the Lagrangian of the variational approach, we recover all known invariants for the statics and dynamics of rods and ribbons, and we extend the approach to vibrations. Finally, we show how the arclength invariant may be used to sometimes obtain pivotal information on elastic rod problems or to test the accuracy of numerical codes.
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