Resumen de Lagrangians of a non-mechanical type for second order Riccati and Abel equations

José Fernando Cariñena Marzo , Manuel F. Rañada

• The Helmholtz approach to the inverse problem of the Lagrangian dynamics is studied first in the particular case of the second-order Riccati equation and then in the case of the second-order Abel equation. The existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class (neither potential nor kinetic term). These second-order Riccati and Abel equations are studied by means of their Darboux polynomials and Jacobi last multipliers. The existence of a family of constants of the motion is also discussed.