Basically, the Lagrangian is where T is kinetic energy and V is potential energy (some systems are easier to describe than others). Right off the bat you’ll notice that has no x or y (just z), so if you change the x or y it doesn’t change at all. It turns out that this gives you conservation of momentum in the x and y directions (that’s not obvious, btw). As a result, changing z changes , and you don’t have conservation of momentum in the z direction (up-down direction). The theorem that describes the correspondence between symmetries and conservation laws is “Noether’s theorem“.

When you want to outsmart the world, you turn to the facts. And the facts are in the science.