Hello I have a short question.
Say I would consider a pendulum and define the Lagrangian as usual being \begin{align} L = \frac{1}{2} m(\dot{x}^2 + \dot{y}^2) - mgy \color{red}{-} \lambda (x^2 + y^2 - \ell^2). \end{align} Then I derive the equations of motion. For the $x$ component I have \begin{align} m\ddot{x} = \color{red}{-}\lambda x \, . \end{align} So at this stage I noticed that a Lagrangian with a term $$\color{red}{+}\lambda(x^2+y^2-\ell^2)$$ would result the same equation with a reversed sign, but as I understand this would mean a tension force in the reversed direction. Is it true that there is only one sign being correct, the minus sign as usual?