Riley hobson bence · Problem 26.28

Numerical methods

Mathematics — Methods · tensors-geodesics-arc-length

Problem

A curve

\mathbf{r}(t)

is parameterised by a scalar variable

t

. Show that the length of the curve between two points,

A

and

B

, is given by

L = \int_A^B \sqrt{g_{ij} \frac{du^i}{dt} \frac{du^j}{dt}}\, dt.

Using the calculus of variations (see chapter 22), show that the curve

\mathbf{r}(t)

that minimises

L

satisfies the equation

\frac{d^2u^i}{dt^2} + \Gamma^i_{jk} \frac{du^j}{dt} \frac{du^k}{dt} = \frac{\ddot{s}}{\dot{s}} \frac{du^i}{dt},

where

s

is the arc length along the curve,

\dot{s} = ds/dt

and

\ddot{s} = d^2s/dt^2

. Hence, show that if the parameter

t

is of the form

t = as + b

, where

a

and

b

are constants, then we recover the equation for a geodesic (26.101). [ A parameter which, like

t

, is the sum of a linear transformation of

s

and a translation is called an affine parameter. ]