Prolongations of Lie groupoids and Lie algebroids

We describe a method of prolonging a Lie groupoid to higher-order groupoids in a way which, unlike the tangent functor, takes account of the double fibration. The usual Lie algebroid may be identified with the identity submanifold of the first prolonged groupoid. The Lie algebroid generated by the first prolonged groupoid is identified with the "prolonged algebroid" used recently in the study of Lagrangian dynamics.