The

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

Planar central configuration estimates in the

n-body problem. Ergodic Theory Dynam.

The coordinate system of the

N-body problem is shown in Figure 1.

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

N-body problem. Large numbers of numerical integrators [3, 4] and ODE solvers for non stiff problems [5, 6, 7] are used to find numerical approximations of differential equations at Eqs.

Since the general solution of the

n-body problem cannot be given, great importance has been attached to search for particular solutions from the very beginning.

The classical

n-body problem [1, 2] concerns the motion of n mass points moving in space according to Newton's law:

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.

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. The text requires no prior familiarity with astronomy or space science, but assumes knowledge of calculus and elementary vector analysis.

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.