# Ex 1.1: Motion on a Sphere, 2

Functional Differential Geometry

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The metric for a unit sphere, expressed in colatitude $\displaystyle{\theta}$ and longitude $\displaystyle{\phi}$, is

$\displaystyle{g(u,v) = d\theta(u) d \theta(v) + (\sin \theta)^2 d \phi (u) d \phi (v)}$

Compute the Lagrange equations for motion of a free particle on the sphere …

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(define ((L2 mass metric) place velocity)
(* 1/2
mass
((metric velocity velocity) place)))

(define ((Lc mass metric coordsys) state)
(let ((x (coordinates state))
(v (velocities state))
(e (coordinate-system->vector-basis
coordsys)))
((L2 mass metric)
((point coordsys) x) (* e v))))

(define the-metric
(literal-metric 'g R2-rect))

(define L
(Lc 'm the-metric R2-rect))

(L (up 't (up 'x 'y) (up 'vx 'vy)))

(show-expression
(L (up 't (up 'x 'y) (up 'v_x 'v_y))))


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$\displaystyle{g(u,v) = d\theta(u) d \theta(v) + (\sin \theta)^2 d \phi (u) d \phi (v)}$

(show-expression
(L (up 't
(up 'theta 'phi)


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When $\displaystyle{R = 1}$ and $\displaystyle{ [g] = \begin{bmatrix} 1 & 0 \\ 0 & (\sin \theta)^2 \end{bmatrix}}$,

$\displaystyle{ \frac{1}{2} m (\sin \theta)^2 \dot \phi^2 + 0 + \frac{1}{2} m \dot \theta^2}$

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— Me@2022-10-27 10:30:50 AM

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