| > How do I compute this function? For the kind of problems in physics 101, the Lagrangian is just kinetic energy minus potential energy: L = T - U. This is definitionally true; there's no "why" to that. Think of L as the formal specification of a system. There's no point in asking "why does this mass on a spring have L = 1/2mv^2 - 1/2kx^2": that equation specifies the problem. In higher level physics, e.g. QFT, there are often identifiably kinetic and potential terms as well. However, there are also often other terms. At that level, you generally write L = sum of the most general possible terms that don't violate symmetries you want the system to have. This ability to work backwards from symmetries to L is to me the more useful perspective on Noether's theorem than going from knowing L to the symmetries. > Do I define q(s) any way I want, or do I solve q(s) given the constraints of the system? q(s) is the function which minimizes the time integral of L, and is solved for via the [principle of least action](https://en.wikipedia.org/wiki/Stationary-action_principle), which requires functional derivatives to understand. Most physicists, myself included, only marginally understand the math here and do a bit of hand waving. "d" "del" "delta" potato potahto. The principle of least action also generates "equations of motion" from L. These are the differential equations that can be solved for x(t). So, in short: a system is formally specified by its L, which has measurable meaning only because of the principle of least action. If you didn't specify the principle of least action, or something else, it would be nearly meaningless to give an expression for L. You might as well say: "the mass on a spring has a potato of 1/3kx^97 - 1/2m^8*v^3 + 3". "Nearly" in that at least that would tell me that two systems with unequal potato are different systems. |