# Multiple solutions to Lambert's problem

I'm solving the Lambert's problem, and already have written a program, which solves the BVP and computes the velocities required to transfer from one point to another in a given time.

However, I discovered, that sometimes it flies in the opposite direction, and therefore requires more Delta-V.

My question is:

• How many solutions (velocities) exist in case of Lambert's problem?
• How to get the solution, which requires less Delta-V?
• I, for one, have learned to ignore users who make repeated edits to their questions. I am far from alone in that regard. You can get lots of help from very knowledgeable members if you follow the SE paradigm. Or you can continue to edit your questions so that answers to the original version are completely out of place. The result of the latter choice is that eventually you will receive no help whatsoever. Your choice. – David Hammen Apr 22 '17 at 6:21

How many solutions (velocities) exist in case of Lambert's problem?

Lambert's problem has an infinite number of solutions. For example, there's typically a solution that involves more that 0° but less than 180° on the transfer orbit, another than involves more than 180° but less than 360° on the transfer orbit, yet another than involves more than 360° but less than 540°, and so on. There are also solutions that involve reversing direction, essentially equivalent to less than 0° on the transfer orbit.

The long transfers (anything requiring more than 360° on the transfer orbit) take too much time, are very touchy, and are hard to find. They're not worth looking at. The solutions that require reversing direction are extremely expensive in terms of delta V. There's no point in looking at those solutions, either.

That leaves two solutions that are worthy of investigation, the "short way" (more than 0° but less than 180° on the transfer orbit) and the "long way" (more than 180° but less than 360° on the transfer orbit). You need to investigate both.

• I found these notes helpful to start, then this and this, just from a quick search. – uhoh Apr 21 '17 at 12:53
• The angle you mentioned is the True Anomaly? How do you find the angles you mentioned? – Tarlan Mammadzada Apr 21 '17 at 17:23
• @TarlanMammadzada -- Yes, it's true anomaly. The multiplicity of solutions means you do not want to use a generic boundary value solver. How to go about solving Lambert's problem is a different question. While there are a number of questions at this site that involve Lambert's problem, there don't appear to be any questions regarding the mechanisms needed to solve it. So ask that as a new question! – David Hammen Apr 21 '17 at 17:43