Point Charges and Polygonal Linkages

Abstract

We investigate the critical points of Coulomb potential of point charges placed at the vertices of a planar polygonal linkage. It is shown that, for a collection of positive charges on a pentagonal linkage, there is a unique critical point in the set of convex configurations which is the point of absolute minimum. This enables us to prove that two controlling charges are sufficient to navigate between any two convex configurations of a pentagonal linkage

Triangle diagram, distance geometry and Symmetries of Feynman Integrals

“…where here α denotes the angle between the a and ∞ faces of the dual tetrahedron. The identities appeared already in equations (9,10) of [30], while identity (2.10) was noted already in theorem 1 of [39]. The generalization of both λ and B 3 to a simplex of arbitrary dimension is given by the Cayley -Menger determinant [40,41] 2 , which applies also to pseudo-Riemannian metrics.…”

On the uniqueness of trapezoidal four-body central configurations

“…This formula is valid when restricting to planar configurations. A generalization of this formula that also works for non planar configurations uses oriented areas and can be found in [18].…”