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In Lagrangian mechanics, Lagrange's equation of the first kind states that

$$ \frac{\partial L}{\partial r_k} - \frac{d}{dt}\frac{\partial L}{\partial \dot{r_k}} + \sum_{i=1}^C \lambda_i \frac{\partial f_i}{\partial r_k} = 0. \tag{1}$$

Here the constraint functions are embedded using the method of Lagrange Multipliers. The method of Lagrange Multipliers is an optimization technique, used to find local minimas/maximas for an objective function given a set of constraints.

This leads me to believe that the part $ \frac{\partial L}{\partial r_k} - \frac{d}{dt}\frac{\partial L}{\partial \dot{r_k}}$ actually comes from applying the gradient to some objective function. But I can not find any information about this function.

So my question is, is there an objective function here and how can one formulate it? Or is my premise flawed? I would be very thankful for any help or pointers in the right direction.

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  1. If the constraints $(f_1,\ldots, f_C)$ are holonomic, then OP's Lagrange equations of first kind (1) follows from an extended action principle $$ S[r,\lambda]~=~~\int \! dt (L + \sum_{i=1}^C\lambda^i f_i). $$

  2. In case of semi-holonomic constraints or dissipative forces, the situation is more complicated, cf. e.g. this & this Phys.SE posts.

  3. However, within the framework of Newtonian mechanics, the Lagrange equations of first kind can always be derived from d'Alembert's principle, cf. e.g. H. Goldstein, Classical mechanics, chapter 1.

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