Hamiltonian delay equations for central configurations

Abstract

We provide in this paper the hamiltonian delay equations of motion for the newtonian n-body problem deduced from the quantum calculus of variations developed in Cresson 2005; Cresson, Frederico, and Torres 2009; Ryckelynck and Smoch 2013, 2014. These equations are brought into the usual lagrangian and hamiltonian formulations of the dynamics and yield sampled functional equations involving generalized derivatives. We investigate especially homographic solutions to these equations that we obtain by solving algebraic systems of equations similar to the classical ones. When the potential forces are homogeneous, homographic solutions to the delayed and to the classical equations may be related through an explicit expansion factor that we provide. Consequently, perturbative equations both in lagrangian and hamiltonian formalisms are deduced.