Central Configurations and Mutual Differences

D.L. Ferrario
2017 Symmetry, Integrability and Geometry: Methods and Applications  
Central configurations are solutions of the equations λ m_jq_j = ∂ U/∂q_j, where U denotes the potential function and each q_j is a point in the d-dimensional Euclidean space E R^d, for j=1,..., n. We show that the vector of the mutual differences q_ij = q_i - q_j satisfies the equation -λ/αq = P_m(Ψ(q)), where P_m is the orthogonal projection over the spaces of 1-cocycles and Ψ(q) = q/|q|^α+2. It is shown that differences q_ij of central configurations are critical points of an analogue of U,
more » ... an analogue of U, defined on the space of 1-cochains in the Euclidean space E, and restricted to the subspace of 1-cocycles. Some generalizations of well known facts follow almost immediately from this approach.
doi:10.3842/sigma.2017.021 fatcat:nvlcpgbembevdaxmkb7gdvzs64