The Lambert’s problem

The Lambert’s problem is the boundary value problem of the two-body problem under the framework of astrodynamics.

Problem statement

The goal is to solve for the orbit of a body being known two of its position vectors \(\vec{r_1}\) and \(\vec{r_2}\), and the time it takes to travel between those, usually denoted by \(\Delta t\). The geometry of the problem is shown in the figure below:


Because the problem states to find an orbit, it is usually included within the Initial Orbit Determination subject. Nevertheless, Lambert’s problem solution has applications in the field of interplanetary maneuvers and orbital rendezvous too.