The N-body problem represents an initial value problem, comprising the system of differential equations (mentioned in section 2) and initial conditions.

Levi-Civita, The n-Body Problem in General Relativity, D.

Burgess, The n-Body Problem, ChiZine Publications, Toronto 2013.

Following this common thread, we deal with both with the classical

N-body problem of Celestial Mechanics, where interactions feature attractive singularities, and competition-diffusion systems, where pattern formation is driven by strongly repulsive forces.

In celestial mechanics, motion of two bodies under the influence of gravitational attraction is the simplest form of

N-body problem.

Abstract: We provide a class of orbits in the curved N-body problem for which no point that could play the role of the centre of mass is fixed or moves uniformly along a geodesic.

Keywords: Curved N-body problem, spaces of constant curvature, differential equations, first integrals.

Other topics include the stationary

n-body problem in general relativity, asymptotic gluing of asymptotically hyperbolic vacuum initial data sets, the global geometry of spacetimes with toroidal or hyperbolic symmetry, and rates of decay for structural damped models with coefficients strictly increasing in time.

There is a new chapter on the Caledonian symmetric

N-body problem.

Section 5 shows some experimental results with an

N-Body problem application developed using the framework and tuned by MATE.

In this textbook he covers tensors, the foundations of differential geometry such as tangent vector space, tensor algebra and Lie derivative and algebra, symplectic geometry, modern mechanics, including the work of Lagrange and Hamilton and perturbation, and the

N-body problem as an original methods for large N in terms of equations and integrals and statistical mechanics.

Newton's

N-body problem studies how heavenly bodies move in settings where the dynamics are dictated by Newton's law of motion.