# How can I show the ideal path from Earth to Mars in a vacuum using calculus?

I would like to use calculus to try and model the optimal path to mars in space, assuming the rocket starts with an initial velocity in the vacuum of space, and only accelerates due to the Sun's gravity. By ideal, I mean the path where the sun has the greatest positive effect on the rocket (bringing it closer to mars). So far, I have set the starting point of the path at the perihelion of earth, in order to maximize the assistive gravitational forces when the rocket initially leaves Earth. I know of Hohmann transfer, and that it is the best path for the rocket, but I need to know how to show this. I have not found any calculus based, or even any proof for the Hohmann transfer. It is hard question to visualize due to its vagueness perhaps, so I am sorry about that.

• Possible hint: remember that any reasonable function can be written as a power series. So, just find the best power series for x(t), y(t), z(t) and figure out what functions they correlate to. You probably know what you're doing is en.wikipedia.org/wiki/Calculus_of_variations Aug 3, 2022 at 16:51
• You seem to be asking us to do your homework for you. We'd like to see more evidence of effort spent trying to find a solution yourself. For example, what function are you trying to optimize, and where did you get it? Aug 3, 2022 at 17:02
• @BarryCarter do you think you could perhaps elaborate a bit more on how I can use it for my problem specifically? I am actually sixteen, and this project is more of a personal interest, however I'd like to treat it as a proper investigation. I looked into calculus of variations and geodesics and I think I understand where your going, and it seems pretty applicable, I guess I just need some help understanding it better. Aug 3, 2022 at 17:18
• To be honest, I don't know that much about it either. Unfortunately, explaining a subject isn't really a good fit for SE. I did get your email and will send you an invite to a Google Space to see if I can help more Aug 4, 2022 at 12:44
• @Wyck I think the OP is looking for a proof or at least evidence of why this is most efficient path Aug 4, 2022 at 14:24