Multi-Platform Optimization of Rail Curve Trajectories: A Variational Approach Using Euler-Lagrange Dynamics and SLSQP Algorithms

The geometric design of curved tracks directly impacts travel time, energy consumption, and safety in high-speed rail systems. This research establishes a comprehensive framework for optimizing curved rail trajectories. By applying the Euler-Lagrange equation and polynomial approximations, we derived a general time-integral equation validated through Python (SLSQP) and MATLAB simulations. Results indicate a 55.65% reduction in theoretical travel time while maintaining rigorous safety and stability constraints via the Stanley Controller model.