Lambert's problem

In celestial mechanics Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, solved by Johann Heinrich Lambert. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination. 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 for which the Kepler orbit is the general solution.

The precise formulation of Lambert's problem is as follows:

Two different times ${\displaystyle \ t_{1}\ ,\ t_{2}\ }$ and two position vectors ${\displaystyle {\bar {r}}_{1}=r_{1}{\hat {r}}_{1},\ {\bar {r}}_{2}=r_{2}{\hat {r}}_{2}\ }$ are given.

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