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A. Ajitha,
- Assistant Professor, Department of Physics, Tagore Institute of Engineering and Technology, Salem, Tamil Nadu, India
Abstract
Symmetry provides a mathematical framework for identifying transformations that preserve the essential structure of a physical or mathematical system. In differential equations, Lie group analysis converts invariance under continuous transformations into a systematic procedure for deriving infinitesimal generators, determining equations, invariant variables, reduced models, and conservation laws. This review examines the foundations and applications of Lie symmetry analysis, with particular emphasis on its relevance to contemporary applied science and engineering. The discussion covers Lie point symmetries, prolongation, determining equations, symmetry reduction, similarity solutions, conservation laws, symmetry classification, nonclassical and generalized symmetries, and symbolic computation. The review further connects these mathematical tools with symmetry in structural and mechanical systems, material science, crystallography, nonlinear dynamics, computational modeling, and symmetry-aware optimization. In structural mechanics, symmetry can reduce computational effort, expose invariant subspaces, clarify repeated natural frequencies, and support vibration and stability analysis. In materials and applied physics, point-group and space-group concepts provide a basis for understanding crystallographic and molecular symmetry, anisotropic properties, and symmetry-controlled phenomena. Computational approaches, including computer algebra, numerical modeling, optimization, and emerging symmetry-aware machine-learning methods, provide new opportunities for automated symmetry detection and model reduction. The review also discusses limitations, including the complexity of determining equations, imperfect or broken symmetry in practical systems, and the distinction between mathematical invariance and complete solvability. By linking classical Lie theory with current applications in engineering, materials, applied physics, and computational science, the paper highlights symmetry analysis as both a theoretical framework and a practical tool for modeling, reduction, and design.
Keywords: Symmetry analysis; Lie groups; Lie point symmetry; differential equations; infinitesimal
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Emerging Trends in Symmetry
| Volume | 02 | |
| 02 | ||
| Received | 23/09/2026 | |
| Accepted | 24/09/2026 | |
| Published | 08/10/2026 | |
| Publication Time | 15 Days |