The constraint forces have the dot product 0 (C⋅x=0, where C and x are vectors, x being the virtual displacement. But the dot product is 0 if C and x are perpendicular. So, are the constraint forces always perpendicular to the virtual displacement? Forces such as tension are not always perpendicular to the virtual displacement. Thus, I am asking why forces like tension are not written in the lagrangian equation of lagrangian mechanics? To be more specific, see the example of the Atwood machine.
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