Gravitational two-body problem
←For further relevant mathematical developments see also Two-body problem, also Kepler orbit, and Kepler problem, and Equation of the center – Analytical expansions
The gravitational two-body problem concerns the motion of two point particles that interact only with each other, due to gravity. This means that influences from any third body are neglected. For approximate results that is often suitable. It also means that the two bodies stay clear of each other, that is, the two do not collide, and one body does not pass through the other's atmosphere. Even if they do, the theory still holds for the part of the orbit where they don't. Apart from these considerations a spherically symmetric body can be approximated by a point mass.
Common examples include the parts of a spaceflight where the spacecraft is not undergoing propulsion and atmospheric effects are negligible, and a single celestial body overwhelmingly dominates the gravitational influence. Other common examples are the orbit of a moon around a planet, and of a planet around a star, and two stars orbiting each other (a binary star).
Two-body problem
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus (although to solve the electron/nucleus 2-body system correctly a quantum mechanical approach must be used).
The two-body problem can be re-formulated as two one-body problems, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the n-body problem for n ≥ 3) cannot be solved in terms of first integrals, except in special cases.
Reduction to two independent, one-body problems
Let x1 and x2 be the vector positions of the two bodies, and m1 and m2 be their masses. The goal is to determine the trajectories x1(t) and x2(t) for all times t, given the initial positions x1(t = 0) and x2(t = 0) and the initial velocities v1(t = 0) and v2(t = 0).
Two-body problem (career)
The two-body problem is a dilemma for life partners (for e.g. spouses or any other couple) in academia, relating to the difficulty of both spouses obtaining jobs at the same university or within a reasonable commuting distance from each other. The central dilemma is thus a no-win situation in which if the couple wishes to stay together one of them may be forced to abandon an academic career, or if both wish to pursue academic careers the relationship may falter due to the spouses being constantly separated. The term "two body problem" has been used in the context of working couples since at least the mid-1990s. It alludes to the insoluble three-body problem in classical mechanics.
More than 70 percent of academic faculty have a working partner, while more than third of faculty have an academic partner.
Solutions
Typical solutions include:
Podcasts:
