The centripetal force law and the equation of motion for a particle on a curved hypersurface

TL;DR

This paper revises the equation of motion for particles on curved hypersurfaces, aligning it with the centripetal force law and emphasizing the importance of geodesics, with implications for quantization of constrained systems.

Contribution

It demonstrates that the extrinsic equation of motion should incorporate geodesic constraints, reformulating it as a centripetal force law applicable in higher dimensions.

Findings

Revised equation aligns with centripetal force law.

Quantization of constrained systems is supported.

Addresses operator form construction issues.

Abstract

It is pointed out that the current form of extrinsic equation of motion for a particle constrained to remain on a hypersurface is in fact a half-finished version for it is established without regard to the fact that the particle can never depart from the geodesics on the surface. Once the fact be taken into consideration, the equation takes that same form as that for centripetal force law, provided that the symbols are re-interpreted so that the law is applicable for higher dimensions. The controversial issue of constructing operator forms of these equations is addressed, and our studies show the quantization of constrained system based on the extrinsic equation of motion is favorable.