Inspectional analysis to search for symmetric solutions. Applications in electromagnetism

• Published in: American journal of physics 2014-12, Vol.82 (12), p.1167-1177
• Main Author: Marrero, José Luis Rodríguez
• Format: Article
• Language: English
• Subjects: Boundary value problems; Dimensional analysis; Electromagnetism; Ordinary differential equations; Partial differential equations; Quantum physics
• ISSN: 0002-9505; 1943-2909
• DOI: 10.1119/1.4891192

A group of symmetries of a mathematical equation is a set of transformations of the dependent and independent variables that leaves the equation invariant. Invariance under symmetry groups is important because it often reduces the set of partial differential equations that describe a problem of continuum physics to a simpler set of ordinary differential equations, which can always be integrated numerically. It is also important because one can make useful inferences about the properties of their solutions without actually solving those equations. Testing differential equations for invariance under groups of transformations is termed inspectional analysis. Examples of the applications of inspectional analysis to electromagnetic problems are discussed. Two groups of symmetries and their associated solutions are studied: the group of scalings with self-similar solutions, and the group of translations with travelling-wave solutions. Also, applications of symmetry groups to boundary value problems are considered. Finally, dimensional analysis, used to reduce the number of variables, parameters, and physical constants in physical problems, is reviewed as a special case of the group of scalings of the fundamental units. It is argued that inspectional analysis is more adequate than dimensional analysis to reduce the number of variables in a differential equation.