- #1

VoxCaelum

- 15

- 0

## Homework Statement

Consider a particle moving on a two dimensional sphere of radius r, whose Lagrangian is given by:

L(q,q')=[itex]\frac{1}{2}[/itex]m[itex]\sum[/itex](q

^{i}')

^{2}

(A) Transform the Lagrangian into spherical coordinates and write down the resulting Euler-Lagrange equations.

(B) Solve the Euler-Lagrange equations corresping to the Lagrangian and write down the solution in the original coordinates (q

^{i}). Express your answer to include the integration constants and denote them as the initial position and velocity.

## Homework Equations

[itex]\frac{∂L}{∂q}[/itex]=[itex]\frac{d}{dt}[/itex][itex]\frac{∂L}{∂q'}[/itex]

## The Attempt at a Solution

I think I have part (A) solved. The problem is part B. I really don't know where to start. But this is what I have for part (A)

mr

^{2}[itex]\phi[/itex]''Sin

^{2}(θ)+[itex]\phi[/itex]'θ'2Sin(θ)Cos(θ)=0

r

^{2}[itex]\phi[/itex]'Sin(θ)Cos(θ)=mr

^{2}θ''

I would not know where to start solving these equations.