Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second-Order Differential Equations

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, t...

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Bibliographic Details
Published in:Journal of applied mathematics 2020, Vol.2020 (2020), p.1-11
Main Authors: Musielak, Zdzislaw E., Rosario-Franco, M., Davachi, N.
Format: Article
Language:English
Subjects:
Algebra
Differential equations
Formalism
Lie groups
Mathematical analysis
Physics
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Summary:A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.
ISSN:1110-757X
1687-0042
DOI:10.1155/2020/3170130