Lie Symmetry Analysis and Conservation Laws of the Axially Loaded Euler Beam

Lili Xia and Xinsheng Ge
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Lili Xia: School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
Xinsheng Ge: Mechanical Electrical Engineering School, Beijing Information Science and Technology University, Beijing 100192, China

Mathematics, 2022, vol. 10, issue 15, 1-13

Abstract: By applying the Lie symmetry method, group-invariant solutions are constructed for axially loaded Euler beams. The corresponding mathematical models of the beams are formulated. After introducing the infinitesimal transformations, the determining equations of Lie symmetry are proposed via Lie point transformations acting on the original equations. The infinitesimal generators of symmetries of the systems are presented with Maple. The corresponding vector fields are given to span the subalgebra of the systems. Conserved vectors are derived by using two methods, namely, the multipliers method and Noether’s theorem. Noether conserved quantities are obtained using the structure equation, satisfied by the gauge functions. The fluxes of the conservation laws could also be proposed with the multipliers. The relations between them are discussed. Furthermore, the original equations of the systems could be transformed into ODEs and the exact explicit solutions are provided.

Keywords: axially loaded Euler beam; Lie symmetry; conservation laws; multiplier method; Noether’s theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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