  This paper develops the notion of implicit Lagrangian systems on Lie
algebroids and a Hamilton--Jacobi theory for this type of system. The Lie
algebroid framework provides a natural generalization of classical tangent
bundle geometry. We define the notion of an implicit Lagrangian system on a Lie
algebroid $E$ using Dirac structures on the Lie algebroid prolongation
$\T^EE^*$. This setting includes degenerate Lagrangian systems with
nonholonomic constraints on Lie algebroids.
