Discrete calculus of variation for homographic configurations in celestial mechanics

摘要

We provide in this paper the discrete equations of motion for the newtonian$n$-body problem deduced from the quantum calculus of variations (Q.C.V.)developed in \cite{Cre,CFT,RS1,RS2}. These equations are brought into the usuallagrangian and hamiltonian formulations of the dynamics and yield sampledfunctional equations involving generalized scale derivatives. We investigateespecially homographic solutions to these equations that we obtain by solvingalgebraic systems of equations similar to the classical ones. When thepotential forces are homogeneous, homographic solutions to the discrete andclassical equations may be related through an explicit expansion factor that weprovide. Consequently, perturbative equations both in lagrangian andhamiltonian formalisms are deduced.