The most fundamental principle in physics is called 'The Principle of Least Action'. It also happens to be a good principle to live by during these trying times
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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
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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.
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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
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