- #1

Yosty22

- 185

- 4

## Homework Statement

Find the Lagrangian and equations of motion for a spherical pendulum

## Homework Equations

L=T-U and Lagrange's Equation

## The Attempt at a Solution

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I found the Lagrangian to be L = 0.5*m*l

^{2}(ω

^{2}+Ω

^{2}sin

^{2}(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 = ml

^{2}Ωsin

^{2}(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(ml

^{2}Ωsin

^{2}(θ)) = ∅*ml

^{2}*sin

^{2}(theta) + 2ml

^{2}Ωsin(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?