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Math Help - geodesic is parametrised by a constant multiple of arc length

  1. #1
    Senior Member slevvio's Avatar
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    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
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  2. #2
    Super Member Rebesques's Avatar
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    The constant is omitted in your notes, though it shouldn't be. Affine transformations of the parameter do not alter geodesy.
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  3. #3
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    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.
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  4. #4
    Super Member Rebesques's Avatar
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    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.
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  5. #5
    Senior Member slevvio's Avatar
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    On revision I return to this topic. What is a closed geodesic in this context?
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