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Yes, Lagrange's mechanics is used widely in practical application. I will get back to that, first some historical remarks. Joseph-Louis Lagrange developed his mechanics, as laid down in the 'Mechanique analytique', by achieving the following two things: -Systematic use of representing physics taking place in terms of interconversion of potential energy and ...

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Okay, two things. 1.) The Lagrangian does not need to be conserved. It is the difference in KE and PE. The sum or total energy is the part that must be conserved. So KE+PE needs to be time invariant, not KE-PE 2.) You can’t assume $x(t)=1-\cos{t}$. The spring force ($-kx$) and Newton’s Second Law ($ma$) must be used to determine the equation of motion. If ...

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I'll try to give a slightly more precise view which might be confusing at first but will pay off later. You are minimizing the action $S[\psi,\psi^*]$ where the square brackets indicate that $S$ is a functional: an object which takes a function as argument and spits out a number. Here $\psi$ and $\psi^*$ are still considered completely independent functions. ...

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I'm not 100% confident in my answer as I am new to this topic, but I hope I am of help. Edit: Your lagrangian is a slightly strange, as pointed out by AccidentalTaylorExpansion, so you may want to double check that it is correct. The below should hold for the lagrangian you have given however. The lagrangian you have is a 'lagrangian density'. If the ...

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