In this talk, we address the study of Hamiltonian chaos considering two different frameworks. The first one concerns the calculation of Lyapunov exponents starting from the called tangent dynamics, which arises from the linearization of Hamilton equations. The second approach is the called geodynamic approach developed by Pettini and collaborators, which is based on the Riemannian reformulation of Hamiltonian dynamics. Our interest will be focused on systems with low dimensionality, such as a system of coupled anharmonic oscillators of the type β-Fermi-Pasta-Ulam Hamiltonian (β -FPU).