Three-body problem

In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. The three-body problem is a special case of the n-body problem. Unlike two-body problems, no closed-form solution exists for all sets of initial conditions, and numerical methods are generally required. In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. The three-body problem is a special case of the n-body problem. Unlike two-body problems, no closed-form solution exists for all sets of initial conditions, and numerical methods are generally required. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun. In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles. The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions r i = ( x i , y i , z i ) {displaystyle mathbf {r_{i}} =(x_{i},y_{i},z_{i})} of three gravitationally interacting bodies with masses m i {displaystyle m_{i}} : where G {displaystyle G} is the gravitational constant. This is a set of 9 second-order differential equations. The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions r i {displaystyle mathbf {r_{i}} } and momenta p i {displaystyle mathbf {p_{i}} } : where H {displaystyle {mathcal {H}}} is the Hamiltonian: In this case H {displaystyle {mathcal {H}}} is simply the total energy of the system, gravitational plus kinetic. In the restricted three-body problem, a body of negligible mass (the 'planetoid') moves under the influence of two massive bodies. Having negligible mass, the planetoid exerts no force on the two massive bodies, which can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.