Question

I had previously worked this problem but after trying toformulate the Lagrange for r and phi, my work became a jumbled messthat took several lines of paper to write down. I thought this wasa problem so I checked Cramster to make sure I was following thecorrect path and found that the Cramster formula for Kinetic Energywas much different from mine. I found the correct answer forPotential Energy, but I do not understand how the Kinetic answerwas formed. The way I originally started the problem was by makinga drawing which was exactly like the one in the answer key andwriting out the conversions from Cartesian to sphericalcoordinates.

x = rsin(α)cos(φ)

y = rsin(α)sin(φ)

z = rcos(α)

The kinetic energy is then

T = (1/2)m[d/dt((rsin(α)cos(φ))^2 + (rsin(α)sin(φ))^2 +(rcos(α))^2)]

Applying the time derivative makes a mess of things and trigidentities are not simplifying things which has led me to theconclusion that something must be wrong in my initial setup. What Ido not understand is how the Cramster answer has a kinetic energyof

T = (1/2)m[rdot^2 + (rsinαφdot)^2]

If an explanation of how this is achieved can be provided, Iwould be incredibly grateful.