A geodesic on a paraboloid
(1)
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(2)
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(3)
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has differential parameters defined by
(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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The geodesic is then given by solving the Euler-Lagrange differential equation
(10)
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As given by Weinstock (1974), the solution simplifies to
(11)
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