Lagrangian equations - ring which is sliding along a wire

[Oomph!]

Homework Statement

Hello. I have this problem:
I have a ring which is sliding along a wire in the shape of a spiral because of gravity.
Spiral (helix) is given as the intersection of two surfaces: x = a*cos(kz), y = a*sin(kz). The gravity field has the z axis direction.
I have to find motion equations and find the wire reaction as a function of time.

2. Homework Equations
I have to solve it with this equation:

The Attempt at a Solution

This is the first time, when I solve a example with Lagrangian equations, so I am not sure what to do.
I created this equations:

I know, it is not the end. I have to find λ, motion equations and the wire reaction. But firstly, please, tell me, if I am right or where is mistake and why. Then I will continue.

Thank you very much.

 

[BvU]

Hi,
can you tell me what
your Lagrangian is ?
your generalized coordinates are ?
all variables and given/known data are, i.e. what your relevant equation symbols represent ?
Then we can continue.

 

[Oomph!]

Do I really need the Lagrangian? When I read the study text, there was Lagrangian only in Lagrange's equations of the second kind, not in Lagrange's equations of the first kind. And I wrote there Lagrange's equations of the first kind, because in my task was, that I have to use this.

My study text is in different language, but the type of equations is same like this on 3rd page: https://www.physast.uga.edu/ag/uploads/2017%20SPRING%20-%20PHYS8011%20-%20HMWK%2004%20-%20Lagrange%20Eqs%20of%201st%20Kind.pdf

Sorry, I am confused, it is new for me.

 

[BvU]

Well, I'm grateful because I had to find out about these first kind Lagrange equations (look suspiciously like Newton's) . At least that revealed what $\Phi$ stands for: the constraint equations. And they are supposed to come in the form $\Phi_\alpha = 0$. Can you make that explicit for me, so we can check your $\partial\Phi_\alpha\over \partial x_j$ ?

Furthermore: I see only one $\lambda$. How many constraints do you have ?

Another tack (problem solving skills): approaching this from the other end: what kind of motion do you expect ? Do your intermediate equations fit that ?