Joseph Klein On variational second order differential equations; polynomial case Abstract: A general study devoted to the inverse problem of the calculus of variations is applied to second order differential equations $\ddot x^k=F^k(x^r,\dot x^r)$, where the $F^k$'s are polynoms of degree two in $\dot x^r$, and conditions are given for the existence of a Lagrangian which is itself a polynom of degree two in $\dot x^r$. Keywords: Geom. of sec. order diff. eq.; calculus of variations. MS classification: 34A55, 70H35.