I'm working on an orbital-class hybrid-propellant rocket design project and stumbled upon this very simple problem: How does one determine the amount of thrust for a multi-stage rocket?
Given the target delta-v, the stage mass distribution which will yield the lowest total vehicle mass can be obtained using the "optimal staging method". However, this does not tell me the size of the engine, e.g., the thrust and the burn time.
Through rigorous literature review, it seems to me that to select the "optimal thrust" for a stage, one needs to look into trajectory design and optimization. Since higher thrust leads to higher aerodynamic drag loss and lower gravity loss (and the opposite for lower thrust), it seems like the optimal thrust would be that which gives us the lowest total delta-v loss. And to determine the total loss, one needs to integrate trajectory into the design process. But since trajectory is a large and complex subject on its own, I'm not sure if I'm in the right direction.
I think you have the right idea: to really get the right answer, you'll need to integrate lots of trajectories. You could just start with typical values, e.g., 1.2-ish for the first stage and less than 1 for the second stage.