Variational formulation of a fully-three-dimensional non-linear beam dynamics by a Lie-group representation

Simone Fiori; Simone Fiori

doi:10.3934/math.2025978

AIMS Mathematics

2025, Volume 10, Issue 9: 21953-21993. doi: 10.3934/math.2025978

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Variational formulation of a fully-three-dimensional non-linear beam dynamics by a Lie-group representation

Simone Fiori ^,

Dipartimento di Ingegneria dell'Informazione, Università Politecnica delle Marche, Via Brecce Bianche, I-60131, Ancona, Italy

Received: 23 July 2025 Revised: 06 September 2025 Accepted: 15 September 2025 Published: 22 September 2025
MSC : 70G45, 70G75

This paper presents a survey of a geometrically exact beam theory formulated within the framework of Lie groups, aimed at providing a mathematically consistent description of slender structures undergoing large displacements and rotations. The beam configuration is modeled as a curve in the space $ {{{\mathbb{SO}(3)}}}\times{{{\mathbb{R}}}}^3 $, enabling a coordinate-free expression of the governing equations. A variational formulation serves as the basis for deriving the equations of motion, which emerge as nonstandard Euler-Lagrange equations on the configuration space. Strain measures arising from the group structure define the internal forces and moments, which couple to the dynamics via balance laws. The formulation automatically incorporates conservation of energy, linear momentum, and angular momentum, and reveals the underlying geometric structure through the appearance of Lie brackets in the angular momentum equation. This framework emphasizes the connection between geometry and mechanics, offering advantages in both physical fidelity and computational stability.
- geometrically exact flexible beam theory,
- inextensible Euler-Bernoulli beam,
- Reissner-Simo beam theory,
- Timoshenko-Ehrenfest beam,
- variational formulation
Citation: Simone Fiori. Variational formulation of a fully-three-dimensional non-linear beam dynamics by a Lie-group representation[J]. AIMS Mathematics, 2025, 10(9): 21953-21993. doi: 10.3934/math.2025978

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Abstract

This paper presents a survey of a geometrically exact beam theory formulated within the framework of Lie groups, aimed at providing a mathematically consistent description of slender structures undergoing large displacements and rotations. The beam configuration is modeled as a curve in the space $ {{{\mathbb{SO}(3)}}}\times{{{\mathbb{R}}}}^3 $, enabling a coordinate-free expression of the governing equations. A variational formulation serves as the basis for deriving the equations of motion, which emerge as nonstandard Euler-Lagrange equations on the configuration space. Strain measures arising from the group structure define the internal forces and moments, which couple to the dynamics via balance laws. The formulation automatically incorporates conservation of energy, linear momentum, and angular momentum, and reveals the underlying geometric structure through the appearance of Lie brackets in the angular momentum equation. This framework emphasizes the connection between geometry and mechanics, offering advantages in both physical fidelity and computational stability.

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