Two-body problem
In classical mechanics, the two-body problem is to calculate and predict the motion of two massive bodies that are orbiting each other in space.
The most prominent example of the classical two-body problem is the gravitational case (see also Kepler problem), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as satellites, planets, and stars.
A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions.
A simpler "one body" model, the "central-force problem", treats one object as the immobile source of a force acting on the other.
One then seeks to predict the motion of the single remaining mobile object.
Such an approximation can give useful results when one object is much more massive than the other (as with a light planet orbiting a heavy star, where the star can be treated as essentially stationary).
However, the one-body approximation is usually unnecessary except as a stepping stone.
For many forces, including gravitational ones, the general version of the two-body problem can be reduced to a pair of one-body problems, allowing it to be solved completely, and giving a solution simple enough to be used effectively.
The two-body problem is interesting in astronomy because pairs of astronomical objects are often moving rapidly in arbitrary directions (so their motions become interesting), widely separated from one another (so they will not collide) and even more widely separated from other objects (so outside influences will be small enough to be ignored safely).
Under the force of gravity, each member of a pair of such objects will orbit their mutual center of mass in an elliptical pattern, unless they are moving fast enough to escape one another entirely, in which case their paths will diverge along other planar conic sections.
If one object is very much heavier than the other, it will move far less than the other with reference to the shared center of mass.
The mutual center of mass may even be inside the larger object.
In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive scalar force field obeying an inverse-square law, with electrostatic attraction being the obvious physical example.
Except perhaps in experimental apparatus or other specialized equipment, we rarely encounter electrostatically interacting objects which are moving fast enough, and in such a direction, as to avoid colliding, and/or which are isolated enough from their surroundings.
[1] Although the two-body model treats the objects as point particles, classical mechanics only apply to systems of macroscopic scale.
Electrons in an atom are sometimes described as "orbiting" its nucleus, following an early conjecture of Niels Bohr (this is the source of the term "orbital").
However, electrons don't actually orbit nuclei in any meaningful sense, and quantum mechanics are necessary for any useful understanding of the electron's real behavior.
Solving the classical two-body problem for an electron orbiting an atomic nucleus is misleading and does not produce many useful insights.
The complete two-body problem can be solved by re-formulating it as two one-body problems: a trivial one and one that involves solving for the motion of one particle in an external potential.
Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently.
be the position of the center of mass (barycenter) of the system.
The force between the two objects, which originates in the two objects, should only be a function of their separation r and not of their absolute positions x1 and x2; otherwise, there would not be translational symmetry, and the laws of physics would have to change from place to place.
Solving the equation for r(t) is the key to the two-body problem.
The solution depends on the specific force between the bodies, which is defined by
as may be verified by substituting the definitions of R and r into the right-hand sides of these two equations.
The motion of two bodies with respect to each other always lies in a plane (in the center of mass frame).
Proof: Defining the linear momentum p and the angular momentum L of the system, with respect to the center of mass, by the equations
where μ is the reduced mass and r is the relative position r2 − r1 (with these written taking the center of mass as the origin, and thus both parallel to r) the rate of change of the angular momentum L equals the net torque N
Introducing the assumption (true of most physical forces, as they obey Newton's strong third law of motion) that the force between two particles acts along the line between their positions, it follows that r × F = 0 and the angular momentum vector L is constant (conserved).
Therefore, the displacement vector r and its velocity v are always in the plane perpendicular to the constant vector L. If the force F(r) is conservative then the system has a potential energy U(r), so the total energy can be written as
