  In this work we present (and encourage the use of) the Williamson theorem and
its consequences in several contexts in physics. We demonstrate this theorem
using only basic concepts of linear algebra and symplectic matrices. As an
immediate application in the context of small oscillations, we show that
applying this theorem reveals the normal-mode coordinates and frequencies of
the system in the Hamiltonian scenario. A modest introduction of the symplectic
formalism in quantum mechanics is presented, useing the theorem to study
quantum normal modes and canonical distributions of thermodynamically stable
systems described by quadratic Hamiltonians. As a last example, a more advanced
topic concerning uncertainty relations is developed to show once more its
utility in a distinct and modern perspective.
