How to find the optimal launch trajectory for a rocket launched from a planet with an atmosphere?

As suggested I have also asked this question on Space Stack Exchange.

As the title states I am trying to find the optimal launch trajectory for a rocket launched from a planet with an atmosphere (Earth) in order to maximise payload to orbit. I have to do a second year physics project and I would very much like to avoid stepping foot in a lab so I am trying to do something more mathematical in nature, and if I am being honest I want to know how to play Kerbal Space Program more optimally.

My background is I am a second year student a university studying theoretical physics (maths & physics pretty much). I have done linear algebra 1 & 2, (Newtonian) mechanics 1 & 2, Calculus 1 & 2, and some physics modules which aren't really relevant. I am currently doing Lagrangian mechanics, real analysis and "equations of mathematical physics" which is Fourier transforms, (more) vector calculus, ODEs and some other stuff. Most of my knowledge of calculus of variations comes from a brief overview of it that I got in my Lagrangian mechanics class. The book for linear algebra module was Algebra by Artin, and for Lagrangian mechanics the recommended books are Landau & Lifshitz volume 1 & 2 to give an idea of the level (I think these 2 will be the most relevant). My level relative to this is that with effort I can get through the homeworks. what I really want from here though is book/papers to read through and some help understanding the maths in them. This is my first time trying to find and read papers so any and all help is appreciated. It is also my first time asking a qustion on maths stack exchange so any critiques are also welcome.

When looking for resources online I had trouble finding anything that answers my question. I have had some luck with NASA's technical reports server and found this paper: Teren, F.; Spurlock, O.F.: Optimal three dimensional launch vehicle trajectories with attitude and attitude rate constraints which looks to go a long way to answering my question but there are still some things which are not clear to me, such as where the equation and constraints that are optimised come from. That appears to come from "Stancil, R.T.; and Kulakowski, L.J.: Rocket Boost Vehicle Mission Optimization. ARS J., vol. 31, no. 7, July 1961, pp 935-942." which is referenced in an earlier paper by Teren and Spurlock that only considers a 2 dimensional vrsion of the problem: Payload optimization of multistage launch vehicles, but I have not been able to find that paper for online for free. I have also come across linear tangent steering which might be worth looking at, perhaps someone could shine a light on this?

I'll be the first to admit that I don't understand everything that is going on in these papers so if I've missed the answer to any of my questions I apologise in advance. First this paper states that it doesn't consider the atmospheric phase other than the booster kick angle as generally one wants to minimise the aerodynamic loads on the rocket which makes perfect sense. Does this mean the paper presumes a gravity turn with the only torque applied being the gravity of the planet? Or does solve it more generally and allow for a launch vehicle that can allow a small pitch over rate? If it allows for a small pitch over rate should I use the maximum allowable? I can't see why the logic used to show that you use the maximum for the optimised parts of the trajectory shouldn't be applied in atmosphere either. I haven't gotten to trying to solve for my initial and final conditions yet but if anyone would like to offer some tips for that, they would be most appreciated. I am familiar with some numerical methods but I have a feeling that I am going to struggle numerically solving this in the end so tips or tricks for that would also be a great help.

If you have read all that without giving up you're a saint, and thank you for taking the time to answer.

• Would space.stackexchange.com be a better place to ask? There's at least one JPL engineer there. – Rodrigo de Azevedo Oct 24 '19 at 21:20
• I wasn't sure, I thought since this is an optimisation problem it would fall in the realm of maths, but if there is a JPL engineer on space.stackexchange.com it may well be worth reposting it over there. Would it be frowned upon to also post it there and leave this question up here too? – Glum_Mathematician Oct 24 '19 at 21:33
• Just as long as you notify in your question that the same question has been posted elsewhere (and crosslink them), it should be fine. – Rodrigo de Azevedo Oct 25 '19 at 5:22
• Thank you for the suggestion. I have had more luck over there. – Glum_Mathematician Oct 25 '19 at 17:08