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)
    (up 'thetadot 'phidot))))

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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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