The article is devoted to the dynamics of mechanical systems, the relationship between classical and quantum equations describing quantum states of microparticles. Taking into account that the conservation of probability density in nonrelativistic quantum mechanics leads to the continuity equation, it is shown that this equation can be derived from the Schrödinger equation, which is the basis of quantum theory. At the same time, a method for deriving the continuity equation from the Hamilton-Jacobi equation, which is considered the main equation of analytical mechanics, is presented. As a proof that the classical model (limit) of the Schrödinger equation is the Hamilton‒Jacobi equation, methods for deriving this equation and the continuity equation from the Schrödinger equation are given. This problem can be realized by assuming that Planck's constant is very small (semiclassical approximation), but in this article the transition from the quantum equation to the classical equations is realized by dividing the action function by a power series. Only the first‒order terms of the series were taken into account in the calculations (first‒order approximation). Although the same result is achieved under the specified restrictions, the second approximation allows one to consider the law of conservation of probability and the number of particles coherently (consistently). The conceptual similarity between conservation of the number of particles and probability invariance can be verified by integrating the equation. It is noted that the equations considered in the article differ in degree and nature from ordinary and independently derived differential equations with linear and nonlinear properties. Any problem in classical mechanics can be solved using the Hamilton‒Jacobi equation, and the role of the Schrödinger equation in the semiclassical approximation is determined. The introduction to the article briefly presents information about these cases, and the results obtained are analyzed and conclusions are drawn.