A Lagrangian description of the higher-order Painlevé equations

摘要

We derive the Lagrangians of the higher-order Painlevé equations using Jacobi’s last multiplier technique. Some of these higher-order differential equations display certain remarkable properties like passing the Painlevé test and satisfy the conditions stated by Juráš [M. Juráš, The inverse problem of the calculus of variations for sixth- and eighth-order scalar ordinary differential equations, Acta Appl. Math. 66 (1) (2001) 25–39], thus allowing for a Lagrangian description.