False constraints. A toy model for studying dynamical systems with degenerate Hessian form

TL;DR

This paper introduces a toy model to analyze the differences between mechanical systems with singular and non-singular Hessians, shedding light on the physical viability of certain geometric models of spinning particles.

Contribution

The paper presents a simplified toy model to study the effects of Hessian degeneracy on the motion of relativistic rotators, clarifying issues related to singular Hessians in physical models.

Findings

Hessian singularity can lead to physically unviable models.

The toy model explains the contradiction between Hessian singularity and unique solutions.

Singular Hessians are identified as a defect in geometric models of spinning particles.

Abstract

This paper studies various aspects of the motion of relativistic rotators, both in the presence and absence of external fields, using a toy model which, in a sense, can be regarded as a non-relativistic limit of the rotators. In a simpler setup, this enables one to gain an insight into the principal difference between mechanical systems with singular and non-singular Hessian, whilst avoiding the complications resulting from the more intricate form of the equations of motion in the fully relativistic regime. In particular, one can comprehend the apparent contradiction between Hessian singularity and simultaneous occurrence of unique solutions for the motion of the fundamental relativistic rotator minimally coupled to the electromagnetic field. With the aid of the toy model the author supports and illustrates his thesis put forward elsewhere that the Hessian singularity is a defect that makes physically unviable some geometric models of spinning particles considered in the literature.