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Paraboloid Geodesic

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A geodesic on a paraboloid

x=sqrt(u)cosv
(1)
y=sqrt(u)sinv
(2)
z=u
(3)

has differential parameters defined by

P=((partialx)/(partialu))^2+((partialy)/(partialu))^2+((partialz)/(partialu))^2
(4)
=1+1/(4u)
(5)
Q=(partialx)/(partialu)(partialx)/(partialv)+(partialy)/(partialu)(partialy)/(partialv)+(partialz)/(partialu)(partialz)/(partialv)
(6)
=1/(2sqrt(u))(cosv-sinv)
(7)
R=((partialx)/(partialv))^2+((partialy)/(partialv))^2+((partialz)/(partialv))^2
(8)
=u.
(9)

The geodesic is then given by solving the Euler-Lagrange differential equation

((partialP)/(partialv)+2v^'(partialQ)/(partialv)+v^('2)(partialR)/(partialv))/(2sqrt(P+2Qv^'+Rv^('2)))-d/(du)((Q+Rv^')/(sqrt(P+2Qv^'+Rv^('2))))=0.
(10)

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

u-c^2=u(1+4c^2)sin^2{v-2cln[k(2sqrt(u-c^2)+sqrt(4u+1))]}.
(11)

See also

Geodesic

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References

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

Referenced on Wolfram|Alpha

Paraboloid Geodesic

Cite this as:

Weisstein, Eric W. "Paraboloid Geodesic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParaboloidGeodesic.html

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