In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination.[1]
Suppose a body under the influence of a central gravitational force is observed to travel from point P1 on its conic trajectory, to a point P2 in a time T. The time of flight is related to other variables by Lambert's theorem, which states:
Stated another way, Lambert's problem is the boundary value problem for the differential equation of the two-body problem when the mass of one body is infinitesimal; this subset of the two-body problem is known as the Kepler orbit.
The precise formulation of Lambert's problem is as follows:
Two different times and two position vectors are given.
Find the solution satisfying the differential equation above for which