Classical Mechanics (Kibble & Berkshire) · Problem 13.14

Hamilton's equations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · geodesics-calculus-of-variations

Problem

Geodesics on a paraboloid Solve the problem of finding the shortest path between two points

P(0, 1, 1)

and

Q(0, -1, 1)

on the surface of the paraboloid

z = x^2 + y^2

. Let

\mathcal{C}

be a path lying in the surface that connects

P

and

Q

. Show that the length of

\mathcal{C}

is given by

L[r] = \int_{-\pi/2}^{\pi/2} \left[r^2 + \left(1 + 4r^2\right)\dot{r}^2\right]^{1/2} d\theta,

where

r = r(\theta)

is the polar equation of the projection of

\mathcal{C}

on to the plane

z = 0

, and

\dot{r}

means

dr/d\theta

. Now find the function

r(\theta)

that minimises

L

. It is easier to work directly with the second order E-L equation (which can be found with computer assistance). Solve the E-L equation numerically with the initial conditions

r(0) = \lambda

\dot{r}(0) = 0

and choose

\lambda

so that the path passes through

P

(and, by symmetry,

Q

). Plot the shortest path using 3D graphics.

Related theory

Calculus Of Variations

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