What is a Lagrangian, and why does nature follow least action?
A Lagrangian is one function of where a system is and how fast it is changing, a single line its whole behavior can be regenerated from. Its running total along a path is the action, and nature takes the path where that total sits level under small nudges, like a valley floor. For a thrown ball the function is energy of motion minus energy of height. Since the level spot can be a peak or saddle, the rule is stationary action. Maupertuis saw a supreme being in it, because it seems to know where the ball lands. No foresight is needed. If each step's odds depend on the present alone, minus the logarithm of a history's odds is a sum of per-step terms, totted up like a receipt. Matching that tally to physical action is separate mathematics.
This answer comes from OPH, Observer Patch Holography. OPH is a rigorous, consistent and complete description of the unified, self-creating computational Universe we inhabit. Its consistency is machine-checked: the fixed point at its core is constructed explicitly, and a theory with a model contains no contradiction.
To understand OPH and check every step yourself, start with the free book Reverse Engineering Reality. It builds the theory from the ground up, and the OPH Notebook answers your questions while you read.
read the free bookask the notebookor ask sage:read chapter 31: What Is a Lagrangian, and Why Is It the Last Thing?watch a universe make its own particles