# Ex 1.1: Motion on a Sphere

Functional Differential Geometry

.

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 and convince yourself that they describe great circles. For example, consider the motion on the equator $\displaystyle{\theta = \pi/2}$ and motion on a line of longitude ($\displaystyle{\phi}$ is constant).

~~~

(define ((Lfree mass) state)
(* 1/2 mass (square (velocity state))))

(define ((sphere->R3 R) state)
(let ((q (coordinate state)))
(let ((theta (ref q 0)) (phi (ref q 1)))
(up (* R (sin theta) (cos phi))
(* R (sin theta) (sin phi))
(* R (cos theta))))))

(define ((F->C F) local)
(up (time local)
(F local)
(+ (((partial 0) F) local)
(* (((partial 1) F) local)
(velocity local)))))

(define (Lsphere m R)
(compose (Lfree m) (F->C (sphere->R3 R))))

(show-expression
((Lsphere 'm 'R)
(up 't
(up 'theta 'phi)

(show-expression
(((Lagrange-equations
(Lsphere 'm 'R))
(up
(literal-function 'theta)
(literal-function 'phi)))
't))


\displaystyle{ \begin{aligned} - \sin \theta (D \phi)^2 \cos \theta + D^2 \theta &= 0 \\ 2 \sin \theta D \theta D \phi \cos \theta + D^2 \phi (\sin \theta)^2 &= 0 \\ \\ D^2 \theta &= (D \phi)^2 \cos \theta \sin \theta \\ D( D \phi (\sin \theta)^2) &= 0 \\ \\ \end{aligned}}

.

So

\displaystyle{ \begin{aligned} D \phi (\sin \theta)^2 &\equiv C \\ \\ \end{aligned}}

for some constant $C$.

.

Since \displaystyle{ \begin{aligned} D \phi (\sin \theta)^2 &= 0 \\ \\ \end{aligned}} for some $\theta$,

\displaystyle{ \begin{aligned} D \phi (\sin \theta)^2 &\equiv 0 \\ \\ \end{aligned}}

This is equivalent to setting up the coordinate system such that the initial value of $\theta$ equals zero.

.

Also, since \displaystyle{ \begin{aligned} (\sin \theta)^2 &\ne 0 \\ \\ \end{aligned}} for some $\theta$,

\displaystyle{ \begin{aligned} D \phi &\equiv 0 \\ \\ \end{aligned}}

.

— Me@2022-10-08 04:56:27 PM

.

.