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Classical and quantum-mechanical corrections to the solution of the Euler–Lagrange equation

V. P. Koshcheev

Moscow Aviation Institute (National Research University), Strela branch, Zhukovsky, Moscow Region, Russian Federation

Abstract: Using the Jacobi equation, we constructed a chain of closed systems of first-order ordinary differential equations that describe the evolution of moments, where the initial conditions can include Planck's constant. We showed that the evolution of moments of different orders occurs independently of each other. We demonstrated that the source of dynamic chaos is, in particular, nonzero initial conditions for the system of equations governing second moments. This is because the mean square fluctuations of position and momentum, which measure the “divergence” of trajectories, grow exponentially fast in the region of negative Gaussian curvature of the potential. Since the solutions of the Jacobi equation must be small corrections to the solution of the Euler–Lagrange equation, the reverse influence of these corrections on the Euler–Lagrange equation can be accounted for using the Monte Carlo method for dynamic variables. This, in turn, leads to the emergence of nonzero initial conditions for higher-order moments.

Keywords: Euler–Lagrange equation, Jacobi equation, chain of closed systems of first-order ordinary differential equations, average values of position and momentum operator correlators, dynamic chaos.