Small oscillations of mechanical systems are considered. The corresponding linear differential equations with constant coefficients are derived using the two methods: the linearization of the initial nonlinear system of equations or preliminary reducing of the expressions for kinetic and potential energies to quadratic forms with constant coefficients. General solutions to equations of small oscillation are found, the notions of natural frequencies and normal modes of oscillation are introduced, their properties are studied. The case of zero frequency and the case when several natural frequencies coincide are examined with the help of normal coordinates. The Rayleigh theorem and the Courant theorem are proved. Free small oscillations in the presence of resistance are considered. The Thomson and Tait theorems on the influence of dissipative and gyroscopic forces on the stability of state of equilibrium are presented. Forced oscillations under the action of arbitrary and periodic forces are considered. The relationship between the impulse transient function and the transfer function is established.