Exploring a Novel Approach for Determining Geodesics on Surfaces of Revolution Using the Generalized Bernoulli Method

In this work, we explore the Generalized Bernoulli Method (GBM) as a systematic approach to determining geodesics on surfaces of revolution. GBM offers an alternative framework for solving variational problems expressed through arc-length integrals or parametric representations, simplifying the complexity of conventional Euler-Lagrange equation-based methods. Our primary objective is to streamline the derivation of differential equations that govern geodesic paths on such surfaces. To demonstrate the effectiveness of GBM, we examine three case studies: a right circular cone, a general surface formed by rotating a curve around the z-axis, and a pseudosphere. These examples illustrate that GBM relies on basic calculus and algebra, providing a simpler and more accessible method compared to traditional techniques. Ultimately, the GBM emerges as a practical and efficient tool for deducing geodesic equations on surfaces of revolution.