Discrete calculus of variation for homographic configurations in celestial mechanics

TL;DR

This paper develops discrete equations of motion for the Newtonian n-body problem using quantum calculus of variations, focusing on homographic solutions and their relation to classical solutions, with explicit expansion factors.

Contribution

It introduces a novel discrete calculus framework for celestial mechanics and derives new equations for homographic solutions, linking them to classical solutions through explicit expansion factors.

Findings

Derived discrete equations of motion from quantum calculus of variations.

Established relations between discrete and classical homographic solutions.

Provided perturbative equations in Lagrangian and Hamiltonian formalisms.

Abstract

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 usual lagrangian and hamiltonian formulations of the dynamics and yield sampled functional equations involving generalized scale 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 discrete and classical equations may be related through an explicit expansion factor that we provide. Consequently, perturbative equations both in lagrangian and hamiltonian formalisms are deduced.