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

Abstract

In the present paper we apply geometric methods, and in particular the reduced energy-momentum (REM) method, to the analysis of stability of planar rotationally invariant relative equilibria of three-point-mass systems. We analyse two examples in detail: equilateral relative equilibria for the three-body problem, and isosceles triatomic molecules. We discuss some open problems to which the method is applicable, including roto-translational motion in the full three-body problem.

Cotangent bundle reduction and Poincaré–Birkhoff normal forms

“…We note that a well established method for determining the stability of reduced systems is the reduced energy-momentum method which was introduced in [36]. For an application to the three-body problem, see also [37]. The reduced energy-momentum method does however not provide a means to compute higher order normal forms as we will do now following Appdenix C. Since we are only interested in demonstrating the basic principle of a normal form computation we will restrict ourselves to the normal of order 4.…”

Phase space structures governing reaction dynamics in rotating molecules

“…A more canonical way of expressing the equations of motion is useful, and we hope that the theory presented in this paper will also be of interest for studies of N -body systems not only from the perspective of reaction dynamics but also from other perspectives. For relevant applications, we refer to [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”