Objective A sparse identification method based on linear multistep methods was proposed for the identification of second - order dynamical systems. Methods Firstly a basis function library containing nearly all possible basis functions was constructed to approximate the right-hand function of the potential second-order dynamical system. Then the second-order dynamical system was discretized using linear multistep methods. Next under the guidance of the generalized least squares principle a suitable approximate covariance matrix of the noise residual term was selected and this matrix was used to weigh the minimization issues obtained by the above process to reduce the influence of noise on the model identification process. Finally a sparse regression algorithm was used to select the most meaningful minimal feature terms from the basis function library and solve their corresponding coefficients through sparse iteration. Results Numerical experiments were done to explore potential second-order dynamical systems using the linear multistep sparse identification method with different time steps and noise levels. These experiments showed that the proposed method had higher accuracy and better robustness in identifying unknown second-order dynamical systems. Conclusion The effectiveness of the algorithm is verified by numerical experiments.