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Paraboloid Geodesic

A Geodesic on a Paraboloid has differential parameters defined by

$\displaystyle P$ $\textstyle \equiv$ $\displaystyle \left({\partial x\over\partial u}\right)^2+\left({\partial y\over\partial u}\right)^2+\left({\partial z\over\partial u}\right)^2$  
  $\textstyle =$ $\displaystyle 1+{\cos^2 v\over 4u}+{\sin^2 v\over 4 u} = 1+{1\over 4u}$ (1)
$\displaystyle Q$ $\textstyle \equiv$ $\displaystyle {\partial^2x\over\partial u\partial v}+{\partial^2y\over\partial u\partial v}+{\partial^2z\over\partial u\partial v}$  
  $\textstyle =$ $\displaystyle 0+u\cos^2 v+u\sin^2 v= u$ (2)
$\displaystyle R$ $\textstyle \equiv$ $\displaystyle 0-{\sin v\over 2\sqrt{u}}+{\cos v\over 2\sqrt{u}}={1\over 2\sqrt{u}}(\cos v-\sin v).$ (3)

The Geodesic is then given by solving the Euler-Lagrange Differential Equation
{{\partial P\over \partial v}+2v'{\partial Q\over \partial v...
...}-{d\over du}\left({Q+Rv'\over \sqrt{P+2Qv'+Rv'^2}}\right)= 0.
\end{displaymath} (4)

As given by Weinstock (1974), the solution simplifies to

\end{displaymath} (5)

See also Geodesic


Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, p. 45, 1974.

© 1996-9 Eric W. Weisstein