Paraboloid Geodesic

A Geodesic on a Paraboloid has differential parameters defined by

$$P \equiv \left({\partial x\over\partial u}\right)^2+\left({\partial y\over\partial u}\right)^2+\left({\partial z\over\partial u}\right)^2$$

$$= 1+{\cos^2 v\over 4u}+{\sin^2 v\over 4 u} = 1+{1\over 4u}$$ (1)

$$Q \equiv {\partial^2x\over\partial u\partial v}+{\partial^2y\over\partial u\partial v}+{\partial^2z\over\partial u\partial v}$$

$$= 0+u\cos^2 v+u\sin^2 v= u$$ (2)

$$R \equiv 0-{\sin v\over 2\sqrt{u}}+{\cos v\over 2\sqrt{u}}={1\over 2\sqrt{u}}(\cos v-\sin v).$$ (3)

The Geodesic is then given by solving the Euler-Lagrange Differential Equation

$${{\partial P\over \partial v}+2v'{\partial Q\over \partial v... ...}-{d\over du}\left({Q+Rv'\over \sqrt{P+2Qv'+Rv'^2}}\right)= 0.$$ (4)

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

$$u-c^2=u(1+4c^2)\sin^2\{v-2c\ln[k(2\sqrt{u-c^2}+\sqrt{4u+1}\,)]\}.$$ (5)

See also Geodesic

References

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