Goodbye Christoffel symbols: A flexible and practical approach for solving physical problems in curved spaces

Abstract

Traditional methods for solving physical equations in curved spaces, particularly in areas like fluid dynamics and continuum mechanics, often face significant complexity due to the necessity of incorporating Christoffel symbols to account for spatial curvature. These symbols complicate the formulation and numerical implementation. In this paper, we present a novel and flexible methodology that entirely obviates the need for Christoffel symbols by fundamentally changing the approach to problem formulation and solution. The method operates by formulating the physical problem directly within a Euclidean 3D Cartesian space, where differential operators are standard and well-defined. The core of our innovation lies in the combined and systematic application of symbolic calculus to perform both the necessary chain rule transformations between the physical curved space and the embedding Euclidean space, and the subsequent projection operations. This powerful symbolic framework allows us to effectively derive and solve the governing equations on the curved geometry without explicitly computing or using Christoffel symbols or specialized curved-space operators. We demonstrate the robustness, flexibility, and advantages of this approach through several examples, including the derivation of the Navier-Stokes equations in cylindrical coordinates, the modelling of complex flows in bent cylindrical tubes, and the simulation of the breakup of viscoelastic threads. These examples highlight the method's ability to simplify the mathematical formulation and provide a robust framework for complex or evolving geometries. The flexibility in choosing basis representations within the Euclidean space is also shown to offer potential benefits for numerical stability in certain applications.