In a vacuum and in a uniform gravitational field, it can be shown that a linear-tangent steering law is optimal
I would like to discuss here the trajectories of Launch Vehicles. In particular first stage ones.As a start, let's assume that :(1) thrust is constant all the way up(2) trajectory maximizes payload mass(3) atmospheric density is modelled as an exponential function (no atmosphere as a zero order approximation)(4) Cx remains constant in sub, trans and supersonic velocity regimes(5) no second order effects like aerodynamic lift, torsional forces, etc. etc.(6) ADDED: payload is inserted into circular orbit (tangential flight for a flat earth approximation)(a) given these constraints, is any analytical solution, even approximated, known to the Lagrangian equations?(b) in particular: for a SSTO LV in vacuum (imagine the LM lifting-off from the Moon) is an analytical solution known that maximize payload from lift-off to orbit?Please note I know almost nothing about this but have a decent math background.Even a reference to a good book would be fine. Thanks in advance for any hint.
I researched for years about different guidance systems and this was the best I could come up with for my simulations. My research showed that most vehicles use low AOA during S-I atmospheric translation and then a closed loop during SII for accurate insertion.