The Army and Navy/Marine Corps use this phrase or something very similar:
“On the preparatory command, shift the weight of the body to the right foot without noticeable movement.”
Nothing like being vague, but in all honesty, marching for each service only facilitates orderly travel from Point A to Point B and the services’ manuals don’t have to be exact. Having said that, if you’re reading this blog, you want to be exact. So what about shifting your weight? Does that facilitate forward movement? Yeah, kinda.
There is a Better, More Specific Way!
The mechanics of marching do not require shifting weight from side-to-side (unless flanking, and that’s about momentum as well). Marching, and specifically, the first step, require weight distribution from the center of the body, forward. So your weight should shift forward from center when given the command of execution.
Read my article, First Step- How to Step Off, to get a complete idea of exactly how you need to begin marching.