1

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?

Qmechanic
  • 201,751
Don Al
  • 1,062
  • Does this answer your question? Point of Lagrange multipliers – tryst with freedom Apr 26 '23 at 03:02
  • No because my question instead is as follows. As every mechanics textbook states, one of the reasons for introducing the Lagrangian formulation is to eliminate the forces of constraint from the equations of motion, so if one is interested on them, why introducing Lagrange multipliers and not simply revert to the Newtonian formulation? – Don Al Apr 26 '23 at 03:08
  • 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 – tryst with freedom Apr 26 '23 at 03:12
  • I fail to see how your comment answers my question. We can make the basic EL equations work by introducing a set of generalised coordinates such that the constraint equations become an identity, as is done, for example, in the case of the simple pendulum. – Don Al Apr 26 '23 at 03:26
  • That's a third way of doing it. The point is, all of them are equivalent @Don Al – tryst with freedom Apr 26 '23 at 03:27
  • @TrystwithFreedom But the essence of my question is that if the Lagrangian formulation was introduced to eliminate constraint forces and we can choose the other two approaches mentioned, and if one is interested in the constraint forces, why not simply reverting to Newtonian mechanics? – Don Al Apr 26 '23 at 03:34
  • 1
    Newton=Lagrange basically. So, it doesn't matter. You culd maybe ask, why use lagrangian at all? Maybe some problem is easier because of it so – tryst with freedom Apr 26 '23 at 03:37
  • For a toy problem, check this. There they substitute variables, but you can try in other ways too – tryst with freedom Apr 26 '23 at 03:40
  • 2
    It is often the case that one wants to use symmetry adapted coördinates, which often eliminates the constraints and it is thus easy to get solutions in the Lagrangian scheme. After getting some simple solutions, one might often want to take a step back and study the constraint forces. That is where the Lagrange multipliers come in. Minimal changes, so you can verify and identify the simple solutions. If you do Newtonian, you often have to work pretty hard just to get an equivalence between the Lagrangian solution and the Newtonian one. People routinely get stuck. – naturallyInconsistent Apr 26 '23 at 05:22
  • Possible duplicates: https://physics.stackexchange.com/q/290879/2451 and links therein. – Qmechanic Apr 26 '23 at 07:10

0 Answers0