〈 Berger | Dillon 〉

but that is a bit circular. in the end, you choose the lagrangian that gives you the right equation.

you're missing the point .. you choose the *general* form of something called a lagrangian and assert that nature must respect the trajectories that make the action stationary .. you need to make definitions some how, right? defining something is not circular

the lagrangian is just a function of coordinates and its derivatives; it can be anything. for me, it is just another way to encode equations.

but it CANT be anything, physically. it must be the kinetic minus potential or else the thing your making stationary has no correlation with reality

like equations cannot be anything for them to match certain physical phenomena. the kinetic minus potential is just another way to express relations between functions and its derivatives

But doesn’t it need to be THAT PARTICULAR relationship between a function and it’s derivative for it to be useful?

indeed, up to a gauge transformation

which i think like another way to say that the underlying equations are "related" to a certain geometry.