Holonomy reduced dynamics of triatomic molecular systems

• Published in: arXiv.org 2010-09
• Main Authors: Çiftçi, Ünver, Waalkens, Holger
• Format: Article
• Language: English
• Subjects: Angular momentum; Bundling; Charged particles; Configurations; Reduction; Symmetry; Three body problem; Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1009.0641

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.