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.

在耗散的陀螺仪系统中，具有四个自由度，并且张量变量以对数（右上指数）和协变（右下指数）形式存在，使用第二种方法获得拉格朗日耗散模型，即{ L，D }模型。阶线性微分方程。使用系统的{ L，D }模型确定广义元素。当满足勒让德变换的先决条件时，便找到了哈密顿量。Lyapunov函数作为残差能量函数（REF）获得。REF由哈密顿量和损耗或耗散能量（为负）之和组成，可通过Lyapunov的第二种方法用于稳定。稳定性条件已经过数学证明。