# Variational calculus problem

I've been wondering lately about a problem that comes from, among other places, an old video game "Lunar Lander". In the game there is a spaceship that has a small tank of fuel, and you're supposed to pilot the ship from an initial height down to the surface of the moon. You win if the landing is sufficiently soft. Your only control (at least in a one-dimensional version of the game) is a dial to control the fuel for thrust during the descent.

Finding the optimal strategy is seemingly a problem in the calculus of variations---out of all possible ways to spend the thrust during the descent, you want the one that minimizes the speed when the lander reaches the surface. Besides the game, given that we've landed real ships on the moon or Mars, I'm guessing this optimization problem is well known. Could someone point me to results in textbooks or the literature?

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This is probably not even that hard to set up yourself, as long as you know all the video game physics involved. – Danu Feb 27 '14 at 11:40
Hi Rus May. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. – Qmechanic Feb 27 '14 at 11:49
@Danu: Right, the set-up isn't too bad. You can start with the ideal (Tsiolkovsky's) rocket equation, add a gravity term and then plug into the machinery of the calculus of variations. The problem I'm running into is that endpoints of the integral for the speed at landing aren't fixed, as they are in many variational problems. That's why I'd like to compare my computation with someone else's. – Rus May Feb 27 '14 at 12:07
An open-ended question of the form "what are possible variational descriptions of rocket landings?" is much too broad for this site. If you show the formalism you're working on, we'll be much better able to help you find solutions, or shortcomings of your model if there are any. – Emilio Pisanty Feb 27 '14 at 13:23
That is really too broad; the answer is essentially "half of rocket science". You should indicate how the answers you expect should be narrower than that to help you get meaningful answers. – Emilio Pisanty Feb 27 '14 at 13:51

## 1 Answer

I think this problem is known in the literature as 'soft landing'. A nice article which may interest you is (1) and the references therein.

(1): Liu, Xing-Long, Guang-Ren Duan, and Kok-Lay Teo. "Optimal soft landing control for moon lander." Automatica 44.4 (2008): 1097-1103.

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This is exactly what I was hoping for. I was unaware of the term "lunar soft landing", but now I see there is a lot of recent literature on the subject. Your reference and the references therein will be extremely useful to me. Thank you much. – Rus May Feb 27 '14 at 18:55