I can't seem to find my mistake in this problem and I think it stems from not understanding how to correctly form constraints and the meaning behind the Lagrangian multiplier.

So first of all I learned in class a nice problem and how to solve it using Holonomic Constraints.

> Q. Consider a mass starting at the top of a quarter of a circle and sliding down, at what point $(x,y)$ will it detach? 
note: as a 2D problem

we solved this by using the constraint as follows:

a. Formulating the constraint - $$x^2 +y^2 - R^2 = 0 $$
b. Writing the equations of motion using the constrained Lagrangian - 
$$L' = \frac{1}{2}m(\dot x^2+\dot y^2) - mgy - \lambda(x^2 +y^2 - R^2)$$
c. Moving over to polar coordinates -
$$x=Rcos(\theta)$$
$$y=Rsin(\theta)$$
d. Substituting this into the Lagrangian

d.1.Using energy conservation we have


$$(\dot\theta)^2= \frac{2g}{R}(1-sin(\theta))$$
d.2. When the mass detaches the constraint force vanishes so $$\lambda = 0$$
e. We plug all those things into our equation of motion and get $$sin(\theta) = 2/3$$


**My Questions are:**

**1**.In my homework assignment I have the exact same problem but the curve is now defined :


$$y(x) = 2 - cosh(x)$$

I worked in a similar manner:

a. $$L' = \frac{1}{2}m(\dot x^2+\dot y^2) - mgy - \lambda(y+cosh(x)-2)$$
b1. $$\dot y = -sinh(x) \dot x$$
b2. $$\ddot y = -(\ddot x sinh(x) + \dot x^2 cosh(x))$$
c. Wrote E-L equations for x and y substituted b2 into y E-L equation, some algebra, and got to this equation :

d.$$\lambda = \frac{m \dot x^2 cosh(x) - mg}{cosh^2(x)}$$

e. Used conservation of energy to get:

$$\dot x^2 = \frac{2g(cosh(x)-1)}{cosh^2(x)}$$ 


Solving this I get that cosh(x) = 2 at the detachment point, basically meaning the mass never detaches and follows the whole curve to the bottom which is impossible.
I think it's because I force lambda = 0 at the detachment point.
I don't understand if it's wrong or not, judging by my lecture notes and the way I understand it, the mass will detach when the constraint force is zero meaning lambda = 0.

**If lambda is indeed not equal to zero (at the detachment point) in this case, why is it and how should I find it's value ?**

**2.What is the point of calculating the partial derivative :**

$$\frac{\partial L'}{\partial \lambda}$$ 
if it always just returns the constraint that we wrote by looking at the system and it's properties