Equations of motion

Abstract

The main purpose of this chapter is to derive the equations of motion of the curved N-body problem on the 3-dimensional manifolds ${\text{M}}_{k}^{3}$. To achieve this goal, we will define the curved potential function, which also represents the potential of the particle system, introduce and apply Euler’s formula for homogeneous functions to the curved potential function, describe the variational method of constrained Lagrangian dynamics, and write down the Euler–Lagrange equations with constraints. After deriving the equations of motion of the curved N-body problem, we will prove that their study can be reduced, by suitable coordinate and time-rescaling transformations, to the unit manifold ${\text{M}}^{3}$. Finally, we will show that the equations of motion can be put in Hamiltonian form and will find their first integrals.