  By using the Hamilton-Jacobi [HJ] framework the topological theories
associated with Euler and Pontryagin classes are analyzed. We report the
construction of a fundamental $HJ$ differential where the characteristic
equations and the symmetries of the theory are identified. Moreover, we work in
both theories with the same phase space variables and we show that in spite of
Pontryagin and Euler classes share the same equations of motion their
symmetries are different. In addition, we report all HJ Hamiltonians and we
compare our results with other formulations reported in the literature.
