Journal of Lie Theory

Our purpose is to introduce by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid a class of Lie groupoids. In other words, we show that the orbits of the co-adjoint representation on the isotropy Lie algebroid of a Lie groupoid are Lie groupoid. We will call this type of Lie groupoid, co-adjoint Lie groupoid. Also, we try to construct and define Hamiltonian systems on the co-adjoint Lie groupoids. By considering the trivial Lie groupoid as an example, we show that our construction can be considered as a generalization of the construction of the Lie groups to the Lie groupoids. Finally we present the types I and II of Hamilton-Jacobi theorem of the Hamiltonian system corresponding to the co-adjoint Lie algebroid.