[Work check] Parametric frictionless wire

Zinggy
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Homework Statement
A friction less wire is wound into a shape described by the parametric equations:
x = acosαλ, y = asinαλ, z = bλ where zis the vertical axis and a, b, and α are positive
constants.

Find the Lagrangian of the bead in terms of λ
using Lagrangian equations of motion, find and expression for λ d^2/dt^2
Relevant Equations
x = acosαλ, y = asinαλ, z = bλ
for my formatting, (dot) implies a single time derivative with respect to the variable

Kinetic Energy = T = (1/2) m (x(dot)2 +y(dot)2 + z(dot)2

Plug in respective values for x y and z -> T= (1/2) m (a2 α2sin2(αλ) λ(dot) +a2 α2cos2(αλ) λ(dot) + b2λ(dot)

After canceling out Sin and cos -> (1/2)m (a2 α2 + b2)λ(dot)2 = Kinetic Energy = T

Potential Energy = U = mg(z - z0)

Plug in value for z U = mgb(λ-λ0)

So the full Lagrangian is L= T-U = (1/2)m (a2 α2 + b2)λ(dot)2 - mgb(λ-λ0)Part 2) Using the equation d/dt(dL/dλ(dot))-dL/dλ = 0

Take necessary derivatives and plug in -> d/dt[m(a2 α2+b2)λ(dot)]+mgb
->m(a2 α2+b2)λ(double dot)+mgb = 0
Do the Algebra to solve for λ(double dot). -> λ(double dot) = -gb/(a2 α2+b2)

Terribly sorry for the formatting on the dots, couldn't figure out how to do that.

Thanks for your time, please point out anything I've done wrong!
 
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If anyone could take the time to check my work I would appreciate it!
 
Assignment is due tomorrow, would really appreciate another set of eyes!
 
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