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I = \int \left( L + \sum\limits_{\alpha=1}^m \lambda_{\alpha} f_a \right) dt.

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f is supposed to be an equation of constraint. I do not understand the significance of the new term. It kind of seems to take the form of a generalized force, but I have not quite been able to convince myself of this. Why is this new term being added? Why is it that solving for lambda gives you the constraint force. And how exactly does one take a derivative of a constraint equation? For instance, say your constraint equation is [itex]r \theta = x[/itex]. This seems like a holonomic constraint if x and theta are you generalized coordinates. Now if I take the derivative with respect to x I should get 1 on one side and 0 on the other. I'm a bit confused by this. Any help would be appreciated. Thanks.