Classical and Lagrangian mechanics in ray tracing: An optimizable framework for inhomogeneous media

Ray tracing is crucial for analyzing wave behaviors in diverse applications. This study presents an approach that extends beyond traditional methods, which typically involve solving second–order differential equations to determine ray trajectories. Leveraging classical momentum–impulse relations and the system’s Lagrangian, we establish a set of intuitive first integrals that circumvent the need for direct differential solutions, paving the way for optimization techniques. Our method employs the “shooting method” a technique for approximating solutions to differential equations by iteratively applying initial conditions. We introduce a novel momentum cost function that streamlines angle determination in anisotropic environments, a significant departure from conventional practices. Numerical validations demonstrate the robustness of our approach, confirming its efficacy in handling both sharp and gradual refractive index changes with high accuracy. The results also highlight the computational intensity required in anisotropic conditions, suggesting potential areas for efficiency improvements. This groundwork not only enhances current understanding but also opens avenues for future research into more complex media.