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HW4_prob2_ConeGeodesic

# HW4_prob2_ConeGeodesic - Cone Geodesic Problem(a Find the...

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Cone Geodesic Problem (a) Find the geodesic curve: the shortest path between two points. In cylindrical coordinates ( ρ, φ, z ), a di ff erential element of length in three dimensions is ds = p ( ) 2 + ( ρdφ ) 2 + ( dz ) 2 , (1) where ρ is the radial distance measured from the z axis ( ρ 2 = x 2 + y 2 ) and φ is the angular displacement measured from the x axis in the x y plane. For displacements constrained to lie on the surface of a cone, ρ = z tan α and d ρ = dz tan α , where α is the half-angle of the cone. The total path length between two points 1 and 2 on the cylinder may, therefore, be written as S = Z 2 1 ds = Z 2 1 r 2 + z 2 · φ 2 dz Z 2 1 f ( z, · φ ) dz, (2) where z is chosen as the independent variable, let tan α = 1 , and the dot · indicates a total derivative with respect to z . The integrand of Eq.(2) does not depend on φ . This is one of the special cases we considered in class, and we know immediately that ∂f/∂ · φ is a constant, say a . So we can write ∂f · φ = a = z 2 · φ r 2 + z 2 · φ 2 . (3)

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HW4_prob2_ConeGeodesic - Cone Geodesic Problem(a Find the...

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