In a dissipative gyroscopic system with four degrees of freedom and tensorial variables in contravariant (right upper index) and covariant (right lower index) forms, a Lagrangian-dissipative model, i.e., {L, D}-model, is obtained using second-order linear differential equations. The generalized elements are determined using the {L, D}-model of the system. When the prerequisite of a Legendre transform is fulfilled, the Hamiltonian is found. The Lyapunov function is obtained as a residual energy function (REF). The REF consists of the sum of Hamiltonian and losses or dissipative energies (which are negative), and can be used for stability by Lyapunov’s second method. Stability conditions are mathematically proven.