Constrained Hamiltonian Systems and Groebner Bases
Vladimir P. Gerdt, Soso A. Gogilidze
TL;DR
The paper addresses the problem of algorithmically determining all constraints in finite-dimensional polynomial constrained Hamiltonian systems and classifying them into first and second class. It advances this goal by marrying Dirac's constraint analysis with Groebner bases, yielding the Dirac-Gröbner algorithm and a Maple implementation that makes the entire procedure algorithmic. Key contributions include a radical-ideal formulation, an extension to non-radical ideals, explicit steps for primary and secondary constraints, and a practical separation of constraints using a Poisson-bracket matrix modulo a Gröbner basis. The authors validate the approach with multiple physics and mechanics examples, achieving fast computations on modest hardware and demonstrating scalability to more complex gauge theories. This work provides a robust symbolic-numeric pipeline for systematic constraint analysis in polynomial dynamical systems, aiding quantization and modeling tasks.
Abstract
In this paper we consider finite-dimensional constrained Hamiltonian systems of polynomial type. In order to compute the complete set of constraints and separate them into the first and second classes we apply the modern algorithmic methods of commutative algebra based on the use of Groebner bases. As it is shown, this makes the classical Dirac method fully algorithmic. The underlying algorithm implemented in Maple is presented and some illustrative examples are given.