Applications of Lie symmetries in beams theory

Abstract

Sophus Lie showed in his work the relation between continuous symmetries and differential equations. Particularly, the application of these symmetries transformations on the dependent and independent variables can obtain new variables to become easier the procedure of integration of the differential equations or reduce the order by extracting the first integrals. One of the great difficulties is to find all the symmetries involved with the differential equation of interest and all existing conservation laws, not just the trivials. Sophus Lie's work was seminal for proposing a powerful method for revealing symmetries and invariants. Unfortunately, the Lie symmetries are not popular and not currently used in engineering problems, even the work of Lie being consolidated in the mathematical community and to have more than a century of existence. However, the use of the theory developed by Lie can be very useful, not only to explain what is done in various problems of engineering involving differential equations, but to give us a clear interpretation of the geometric relation between integrability, that is the possibility or not of a solution by known functions, with the existence of invariants, quantities being conserved. Thus, this research proposal aims to introduce these concepts to the scholarship student by showing the use of Lie symmetries directly applied to reveal all the symmetries and transformations that can be made directly into classical problems involving the Euler-Bernoulli Beam equation. Looking for to apply the Fundamental Lie Theorem to obtain all the invariants associated with all symmetries and, especially, in combinations of these. The challenge and original contribution that this work can generate refers to the proposition of a new method by Lie symmetries to calculate a very useful invariant that is associated with the equation of the beam: the calculation of support reactions in hyperstatic cases and an alternative form for the calculation of the strain energy, for example. (AU)