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SIGMA 8 (2012), 035, 9 pages arXiv:1206.3005
https://doi.org/10.3842/SIGMA.2012.035
A Note on the First Integrals of Vector Fields with Integrating Factors and Normalizers
Jaume Llibre a and Daniel Peralta-Salas b
a) Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
b) Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain
Received February 16, 2012, in final form June 12, 2012; Published online June 14, 2012
Abstract
We prove a sufficient condition for the existence of explicit first
integrals for vector fields which admit an integrating factor. This
theorem recovers and extends previous results in the literature on
the integrability of vector fields which are volume preserving and
possess nontrivial normalizers. Our approach is geometric and
coordinate-free and hence it works on any smooth orientable
manifold.
Key words:
first integral; vector field; integrating factor; normalizer.
pdf (323 kb)
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