Lagrangian neural networks for nonholonomic mechanics

The laws of motion of a Lagrangian system are determined by the principle of stationary action, also known as Hamilton's principle. This principle states that the action is minimal (or stationary) throughout a mechanical process. From this statement, the differential equations known as Euler-Lagrange equations are derived. If the Lagrangian function of a given mechanical system is known, then Euler-Lagrange equations establish the relationship between accelerations, velocities, and positions; that is, the system dynamics are obtained from Euler-Lagrange equations. Hence, the goal of Lagrangian mechanics is to write an analytic expression for the Lagrangian function in appropriate generalized coordinates and then develop the Euler-Lagrange equations symbolically into a system of second-order differential equations whose solutions give the system's trajectory. In many cases, even when Euler-Lagrange equations are available, the solutions are not provided in analytical or explicit forms.