Holonomy reduced dynamics of triatomic molecular systems

TL;DR

This paper introduces a holonomy reduction approach to analyze the rotational symmetry in triatomic molecular systems, revealing a new reduced configuration space and connecting molecular dynamics to charged particle motion in magnetic fields.

Contribution

It develops a holonomy reduction framework for triatomic molecules, providing a novel perspective on their symmetry reduction and dynamics compared to traditional methods.

Findings

Reduced configuration space topology: R_+^3 x S^1

Phase space remains a cotangent bundle after reduction

Connection established with charged particle in magnetic field

Abstract

Whereas it is easy to reduce the translational symmetry of a molecular system by using, e.g., Jacobi coordinates the situation is much more involved for the rotational symmetry. In this paper we address the latter problem using {\it holonomy reduction}. We suggest that the configuration space may be considered as the reduced holonomy bundle with a connection induced by the mechanical connection. Using the fact that for the special case of the three-body problem, the holonomy group is SO(2) (as opposed to SO(3) like in systems with more than three bodies) we obtain a holonomy reduced configuration space of topology $ \mathbf{R}_+^3 \times S^1$. The dynamics then takes place on the cotangent bundle over the holonomy reduced configuration space. On this phase space there is an $S^1$ symmetry action coming from the conserved reduced angular momentum which can be reduced using the standard symplectic reduction method. Using a theorem by Arnold it follows that the resulting symmetry reduced phase space is again a natural mechanical phase space, i.e. a cotangent bundle. This is different from what is obtained from the usual approach where symplectic reduction is used from the outset. This difference is discussed in some detail, and a connection between the reduced dynamics of a triatomic molecule and the motion of a charged particle in a magnetic field is established.