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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 H J 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. • We report the Hamilton–Jacobi [HJ] analysis for Pontryagin and Euler classes. • We report a fundamental HJ differential. • The characteristic equations are reported. • The symmetries of both theories are compared. • The generalized HJ brackets are constructed. [ABSTRACT FROM AUTHOR]