Find the Lagrangian and equations of motion for a spherical pendulum
L=T-U and Lagrange's Equation
The Attempt at a Solution
I found the Lagrangian to be L = 0.5*m*l2(ω2+Ω2sin2(θ)) - mgl*cos(θ) where l is the length of the rod, ω is (theta dot) and Ω is (phi dot). Here, the angle θ is measured vertically down from the z-axis and Φ is measured in the xy-plane.
My question comes when solving the Euler-Lagrange equation for Φ, namely the term: (d/dt)(∂L/∂Ω).
The inner term, ∂L/∂Ω is easy enough: ∂L/∂Ω = ml2Ωsin2(θ). The trick for me is coming when finding the total time derivative of that. I've seen two sources online that give different values, but what I did was:
d/dt(∂L/∂Ω) = d/dt(ml2Ωsin2(θ)) = ∅*ml2*sin2(theta) + 2ml2Ωsin(θ)cos(θ)ω
Here, ∅ = (phi double dot). Is this right? A lot of things I have seen online leave out the ω = (theta dot) factor in the second term. This has to be there for a total time derivative, right?