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First Integrals of Differential Operators from SL (2, ℝ ) Symmetries

Author

Listed:
  • Paola Morando

    (Dipartimento di Scienze Agrarie e Ambientali-Produzione, Territorio, Agroenergia, Università Degli Studi di Milano, 20133 Milano, Italy)

  • Concepción Muriel

    (Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, 11510 Puerto Real, Spain)

  • Adrián Ruiz

    (Departamento de Matemáticas, Escuela de Ingenierías Marina, Náutica y Radioelectrónica, Universidad de Cádiz, 11510 Puerto Real, Spain)

Abstract

The construction of first integrals for S L ( 2 , R ) -invariant n th-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra sl ( 2 , R ) . Firstly, we provide for n = 2 an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of sl ( 2 , R ) , and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for n > 2 , provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of sl ( 2 , R ) is known. Several examples illustrate the procedures.

Suggested Citation

  • Paola Morando & Concepción Muriel & Adrián Ruiz, 2020. "First Integrals of Differential Operators from SL (2, ℝ ) Symmetries," Mathematics, MDPI, vol. 8(12), pages 1-18, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2167-:d:457141
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