Constrained Dynamics on an Ellipse

TL;DR

This work quantizes a particle constrained to move on an ellipse by applying Dirac's method for constrained systems, contrasting it with the circle where a natural angular coordinate yields a simple quantum momentum. The ellipse introduces geometry-dependent Dirac brackets with nonuniform curvature, preventing a straightforward extrapolation from circular results and necessitating a direct operator construction. The authors derive the full set of secondary constraints, compute the constraint matrix, obtain Dirac brackets, and construct explicit self-adjoint momentum operators $\hat{p}_x$ and $\hat{p}_y$ that satisfy the bracket algebra, reducing to the circle case when $a=b$. The study highlights how the curvature profile of the constraint surface shapes the quantum operator structure, providing a complete quantization framework for constrained dynamics on an ellipse.

Abstract

We first review the application of Dirac's method to the dynamics of a classical particle constrained to a circle and its subsequent quantization. Then, we extend the analysis to a particle constrained to move on an ellipse. Particularly, we identify the corresponding Dirac brackets and determine the quantum operators associated with the fundamental dynamical variables.