geodesic is parametrised by a constant multiple of arc length

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December 22nd 2010, 10:53 AM
slevvio

geodesic is parametrised by a constant multiple of arc length

Let $S\in \mathbb{R}^3$ be a regular surface, and $\gamma: I \rightarrow S$ a geodesic, where $I$ is an open interval in $\mathbb{R}$ . We have that $\frac{d}{dt}|\gamma ' (t) | = 0 \implies |\gamma '(t)| = c$ , a constant , i.e.

$<br /> \frac{ds}{dt} = | \gamma' (t) | = c$ ..................(1)

where s is the arc length parameter $s(x) = \int_a^x |\gamma'(p)| dp$ , $a\in I$ . I understand what is happening up to here but then my notes claim that $t=\frac{s}{c}$ . However do we not get that upon rearranging (1)

$ds=cdt \implies s = ct + A$ where A is some constant? Why is A zero? Any help would be appreciated
December 23rd 2010, 01:58 PM
Rebesques

The constant is omitted in your notes, though it shouldn't be. Affine transformations of the parameter do not alter geodesy.
December 27th 2010, 09:47 AM
HallsofIvy

The choice of the added constant, A, is just a choice of where to start measuring the arclength. It is always possible to choose the starting point so that A= 0. I expect that it was left out here because the starting point is not relevant.
December 27th 2010, 03:04 PM
Rebesques

Quote:

it was left out here because the starting point is not relevant

I am not sure about this H, as the geodesic may not be closed.
April 26th 2011, 12:31 PM
slevvio

On revision I return to this topic. What is a closed geodesic in this context?