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

Abstract

It is pointed out that the current form of the 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 this fact is taken into consideration, the equation takes the same form as that for the 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 preferable.

The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere

“…In quantum mechanics, a constrained dynamical system is usually associated with a gauge structure, and it is quite well-understood in, for instance, gravitational field [1,2], condensed matter physics [3,4], quantum fields [5] and particle physics [6]. We are recently interested in the constrained motion of which a particle remains and freely move on a hypersurface [7][8][9][10][11][12][13][14][15][16][17][18], and there are also a lot of papers paying attention to the curvature-induced gauge structure on it [19][20][21][22][23][24][25][26][27][28][29][30][31]. In present article, we demonstrate that the gauge potential arisen from the fundamental algebra, i.e., a generalized angular momentum algebra, on the hypersphere [19] can be a part of the geometric momentum, and the resulting momentum obeys the fundamental quantization conditions.…”

“…where p = Π+ A and Π is the so-called geometric momentum for a spinless particle on the hypersurface is, [7][8][9][10][11][12][13][14][15][16][17][18]…”

No existence of the geometric potential for a Dirac fermion on two-dimensional curved surfaces of revolution

“…Once the motion is relativistically, we have following Dirac brackets containing classical brackets between (x, H) and (p, H) in the following, [27,28], where [f, g] D denotes Dirac bracket for two classical quantities f and g. The meaning of n ∧ [p, H] D = 0 is simple: The free particle on the surface experiences no tangential force. While performing quantization, we have the so-called dynamical quantum conditions (DQCs) [27] accordingly,…”

Heisenberg equation for a nonrelativistic particle on a hypersurface: From the centripetal force to a curvature induced force

“…No matter what form of the surface equation we choose, only the unit normal vector and/or its derivatives enter the physics equation. In classical mechanics, the equation of motion of the particle on the surface is, [1,[13][14][15],…”

“…denotes the classical limit. Clearly, F j (12) goes to zero in classical mechanics for we have an orthogonality n•p = 0, while G j (14) corresponds to the centripetal force −2n j p•∇n•p. The quantities F j and G j can be simplified into, respectively,…”