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My question regards the statement that an equation of motion may be invariant under a Lorentz transformation

I just finished watching the Stanford University special relativity lectures on special relativity, taught by Leonard Susskind.

Throughout the field theory part of the lectures, he teaches us that using invariant quantities such as the proper time, we can form Lagrangians that are invariant under Lorentz transformations and thus the equations of motions derived from the Euler-Lagrange equations will also be invariant.

My question is: Does this mean that a moving observer and a stationary observer will both agree on the equations? I assume the answer is yes but I'm not sure how to get my head around it. If the moving observer measures different lengths and times than the stationary observer, I don't see how they can both agree on the trajectory of a particle or the dynamics of a field or some general system when generally they have these measurement discrepancies.

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Different observers both agree on the equations of motion. They both see the same physics, governed by the same laws.

However, they deal with different initial conditions. One guy sees a stationary object which seems to be moving for another. But equations governing that object are all the same.

Take a very simple example: a particle in flat space. I can say that the equation of motion is just

$$\ddot{\vec{x}}(t) = 0.$$

This means a straight line. So this is what all observers agree about: the particle moves along a straight line. However its velocity is not determined by the equation of motion, but rather by the initial data. This is how observers differ from each other.

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