# Why introduce Lagrange multipliers? [duplicate]

For a non-relativistic particle of mass $$m$$ with a conservative force with potential $$U$$ acting on the particle and a holonomic constraint given by $$f(\mathbf{r},t)=0$$, the system can be incorporated into the Lagrangian formulation via introducing a variable additionally to the coordinates $$\lambda$$, called Lagrange multiplier, with Lagrangian given by $$\mathcal{L}=\frac{1}{2}m|\mathbf{v}|^2-U(\mathbf{r})+\lambda f(\mathbf{r},t),$$ and applying the Euler-Lagrange equation for both the coordinates and $$\lambda$$, the equations of motion are $$m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=-\nabla U+\lambda\nabla f; f(\mathbf{r},t)=0,$$ which is the standard Newton's second law and the force of constraint is identified as $$\mathbf{F}_{\mathrm{c}}=\lambda \nabla f$$.

My question is why bother doing this? I don't have any issues understanding how are they used, I simply want to know the point of them. All mechanics textbooks say that one of the reasons for the introduction of the Lagrangian formulation is so that we can eliminate the need for the constraint force to enter the equations of motion, but here we are reintroducing it after having devised a procedure to eliminate them. Why? If one is interested in the constraint force, why not simply revert to the Newtonian formulation?

• When the variables of the Lagrangian are dependent, then the basic EL equations don't work anymore. For eg, if we have smthn like $L(x , \dot{x}, y , \dot{y})$ and $x(y)$ or something. The point is, we want to generalize the lagrangian formulation to deal with such interdependent variable cases as well (imagine pulley system. For that, we can either substitutes the $y$s in terms of $x$s) or introduce the lagrange multiplier term to make the lagrangrian term still work