Symplectic reduction and topology for applications in classical molecular dynamics

Abstract

This paper aims to introduce readers with backgrounds in classical molecular dynamics to some ideas in geometric mechanics that may be useful. This is done through some simple but specific examples: (i) the separation of the rotational and internal energies in an arbitrarily floppy N-body system and (ii) the reduction of the phase space accompanying the change from the laboratory coordinate system to the center of mass coordinate system relevant to molecular collision dynamics. For the case of two-body molecular systems constrained to a plane, symplectic reduction is employed to demonstrate explicitly the separation of translational, rotational, and internal energies and the corresponding reductions of the phase space describing the dynamics for Hamiltonian systems with symmetry. Further, by examining the topology of the energy-momentum map, a unified treatment is presented of the reduction results for the description of (i) the classical dynamics of rotating and vibrating diatomic molecules, which correspond to bound trajectories and (ii) the classical dynamics of atom-atom collisions, which correspond to scattering trajectories. This provides a framework for the treatment of the dynamics of larger N-body systems, including the dynamics of larger rotating and vibrating polyatomic molecular systems and the dynamics of molecule-molecule collisions.

Stability for Lagrangian relative equilibria of three-point-mass systems

“…Our study continues the theme developed by Marsden and Lin in [LM92], in which the geometrical view of mechanics, and aspects of reduction theory, are presented to a broad intended audience of scientists interested in molecular dynamics. In the present paper we apply geometric methods, and in particular the reduced energy-momentum (REM) method, to the analysis of stability of relative equilibria.…”

“…The atom-atom interaction is governed by laws with the following generic features: (1) they become highly repulsive when the interatomic distance approaches zero (i.e. near collision), (2) attain a finite value (the well depth) when the two atoms orbit circular and uniformly around each other and (3) vanish asymptotically at infinity (see, for instance, [LM92]). A commonly used function is the Morse potential F M D,a,r 0 (r) = D(e −2a(r−r 0 ) − 2 e −a(r−r 0 ) ), where the parameters D and r 0 represent the well depth and the equilibrium distance, respectively, and a controls the width of the potential.…”