I’m in second year at university, eternally grateful for the Lagrangian Framework when it comes to simplifying complex mechanical problems.
However, I don’t really understand how defining the potential of a system, and the boundary conditions of the problem, can return the equations of motion for the situation that the given potential describes.
I know it involves… variations… stationary points… that’s about it.
In: Physics
Disclaimer: this may not be a complete or perfect answer as I also have struggled to get a good intuition for what’s going on behind the math in Lagrangian Mechanics.
What you’re trying to do in a Lagrangian Mechanics problem is to solve a minimization problem, specifically finding the minimum/minima of the action. The action is the integral of the Lagrangian, which in many cases represents the energy of the system while keeping restraints (a pendulum has to move in an arc, a train has to be on the tracks). You can in a roundabout way (though not totally accurately) think of it as finding the funcion/equation that is the minimum energy path for the thing you care about.
The more accurate but less intuitive reason is that the symmetries of action (integral of the Lagrangian) are conservation laws. By using the action, you’re maintaining the math that ensures your system holds to the rules of things like conservation of energy.
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