Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction?
My question arose because I was thinking how could you describe the motion of pendulum with an oscillating suspension point, something like this:
If you take your coordinate frame fixed in the suspension point, you would get the motion of a simple pendulum, which is false (because it's a non-inertial frame and you should have virtual forces).
But if it were moving at constant speed, a moving coordinate frame should give the same equations of motion.
So where does the restriction to non-inertial frames come from in the Lagrangian formalism? I've read that in non-inertial frames the lagrangian would be $L(q,\dot q, \ddot q, \dddot q..., t)$, but I didn't see a detailed explanation.
And I'm not sure if the situation would be similar to Kepler's problem in polar coordinates.
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