doi:10.3934/jgm.2021021engMarrero J.C.de Diego D.M.Martínez E.LOCAL CONVEXITY for SECOND ORDER DIFFERENTIAL EQUATIONS on A LIE ALGEBROIDART-2021-125830A theory of local convexity for a second order differential equation (sode) on a Lie algebroid is developed. The particular case when the sode is homogeneous quadratic is extensively discussed. ©American Institute of Mathematical Sciences2021http://zaguan.unizar.es/record/15114410.3934/jgm.2021021http://zaguan.unizar.es/record/151144oai:zaguan.unizar.es:151144info:eu-repo/grantAgreement/ES/MICINN/PGC2018-098265-B-C31JOURNAL OF GEOMETRIC MECHANICS 13, 3 (2021), 477-499byhttps://creativecommons.org/licenses/by/4.0/deed.esinfo:eu-repo/semantics/openAccess