In this topic,a new approach to the analysis of time-variation dynamics is proposed by use ofLegendre series expansion and Legendre integral operator matrix.The theoretical basis for effective solutionof time-variation dynamics is therefore established,which is beneficial to further research of time-variationscience. In this topic, a new approach to the analysis of time-variation dynamics is proposed by use of Legendre series expansion and Legendre integral operator matrix. The theoretical basis for effective solutionof time-variation dynamics is therefore established, which is beneficial to further research of time -variationscience.