Hamilton-Jacobi Method and Effective Actions of D-brane and M-brane in Supergravity

We show that the effective actions of the D-brane and the M-brane are solutions to the Hamilton-Jacobi (H-J) equations in supergravities. This fact means that these effective actions are on-shell actions in supergravities. These solutions to the H-J equations reproduce the supergravity solutions that represent D-branes in a $B_2$ field, M2 branes and the M2-M5 bound states. The effective actions in these solutions are those of a probe D-brane and a probe M-brane. Our findings imply a relation between the on-shell action in supergravity and the effective action in the dual gauge theory, and can be applied to a study of the holographic renormalization group and a search for new solutions of supergravity.