Functional Differential Geometry

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The metric for a unit sphere, expressed in colatitude and longitude , is

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 and motion on a line of longitude ( is constant).

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(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(sintheta)(cosphi))(* R(sintheta)(sinphi))(* R(costheta))))))(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)(up 'thetadot 'phidot))))(show-expression(((Lagrange-equations(Lsphere 'm 'R))(up(literal-function 'theta)(literal-function 'phi)))'t))

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So

for some constant .

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Since for some ,

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

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Also, since for some ,

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— Me@2022-10-08 04:56:27 PM

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2022.10.09 Sunday (c) All rights reserved by ACHK

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