VARIATIONAL AND INTEGRABLE CONNECTIONS

The two-dimensional symmetric connections whose geodesic equations are derivable from a Lagrangian function are divided into five classes. T his classification is compared with that of Douglas for more general c lasses of systems of ordinary differential equations. Three df the fiv e classes of connections are further investigated and in most cases sp ecific Lagrangians are exhibited. In particular those connections that are engendered by Lagrangians homogeneous in velocities are character ized in terms of the Ricci tensor of each connection. Finally several examples of variational connections that possess integrals of motion a re given, thereby extending the known class of completely integrable s ystems.