Abstract: The research of Lagrange's equations in circuit theory was reviewed. Then, the mutual deductive relationship between Lagrange's equation and Kirchhoff's law were discussed. When deducing Kirchhoff's law by Lagrange's equation, a proper tree for choosing independent generalized coordinates was proposed, and some characteristics of Lagrange's equation in circuit were pointed out. When using Kirchhoff's voltage law (KVL) to deduce Lagrange's equation, there was no need to write the energy function. It was proved that the Lagrange's equations in the circuit were a set of independent KVL equations by taking the network graph theory. This mutual deductive relationship shows that the Lagrange's equation contains Kirchhoff's law, the dynamic behaviors of the circuit also follow the dynamic behaviors of other energy systems, and the circuit and other energy systems are unified at the meaning of Lagrange's equation. Through this mutual deduction, the understanding of the common laws of circuits and other energy systems can be greatly improved. The representative circuit examples in the article can provide direct reference for different application backgrounds.