We generally know that Newton's II Law varies when we change the coordinate system. For example Newton's Law in Cartesian coordinates takes the popular form (F=mx''=mα). But if we express it in polar coordinates, the usual form changes.
Does the same thing happen to Euler-Lagrange equation too? Well, it would be logical to suggest that since the time integral of action takes its minimum (or maximum) value when the system follows the natural trajectory, and Euler-Lagrange equations come from this assumption, then they would stay invariant in any coordinate transformation that describes the physical trajectory. That is what I try to prove in my following analysis:
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