Re: [Phys-l] polar grid navigation

[John Denker <jsd@av8n.com>]

In the context of    http://www.youtube.com/watch?v=rc4r3a6F0aA

On 02/16/2012 10:22 AM, LaMontagne, Bob wrote:
> > Lovely - by the time you did this calculation you would have flown to Chile 

The video describes how a computer would do the calculation, which
is not how a pilot would do it.
 -- A computer could finish the calculation well before you arrived
  in Chile.
 -- A pilot would also finish the calculation in less time than it
  takes to talk about it.  On real-world navigation charts, there are
  strategically-located _compass roses_ ... so all you need to do is
  parallel-transport your vector from a grid-course rose to a suitable
  true-course rose.  Note that different true-course roses will be
  aligned differently, depending on longitude, but this is obvious 
  when you look at the chart and can easily be taken into account.
  This is a good example of analog computing.

Also note that not all south polar grid navigation charts choose the
antimeridian as grid north (as in the video).  For example, see
   http://www.andrill.org/iceberg/blogs/luann/all.php
and search for "Jess".

Also note that in the video, the W and E markings do not refer to 
directions relative to grid north *or* relative to true north.  
Instead they apparently refer to western hemisphere and eastern 
hemisphere.  Indeed, the W corresponds to grid-east and the E 
corresponds to grid-west.  IMHO this is yet another pedagogical 
weakness in the video.


On 02/16/2012 07:25 AM, Rauber, Joel wrote:
> Just a passing thought for those who must teach differential
> geometry, either as part of a math class or a relativity class, or
> just as an interesting feature of coordinate systems, which almost
> all physics classes use.
>
> This is a real-world example of what is known as a "coordinate
> patch".  The collection of all coordinate patches that cover a
> manifold (space) is known as an "atlas".  And we see that if one is
> using lat-longs you are forced to use more than one coordinate patch
> to cover the surface of a two sphere (or in our case a slightly
> deformed two sphere.)

I agree with all that.

We can go even farther down that road.  This is not limited to
relativity or to polar grid navigation.  As I said before, it is
relevant to elementary particle physics and cosmology and *lots*
of stuff in between.

 -- In math, we say that SU(2) is a double covering of SO(3).  This
  applies directly to particle physics since SU(2) describes spin 
  and SO(3) describes ordinary 3-dimensional rotations.  

 -- SU(2) is related to quaternions and spinors and bivectors, which 
  is how one handles attitude in real-world autopilots and flight 
  simulators.  So this is relevant to attitude (orientation) not just
  to navigation (center-of-mass position).

 -- AFAICT the only way to make sense of thermodynamics is to realize
  that things like dT and dS are _vectors_.  This can be taught to
  undergraduates ... which might sound like extra work, but in fact
  is much easier than dealing with the confusion that would otherwise
  result.  This can be formalized using differential topology, but
  you don't need to call it that, and in fact you only need a very
  small watered-down subset of differential topology.  Note that I
  didn't say "differential geometry" since in thermo there is no 
  geometry, i.e. no dot product and no notion of length or angle in 
  thermodynamic state space.

  This is a big deal because it allows students to /visualize/ what
  is going on in thermodynamic state space.  For example, things like
  the cyclic triple derivative rule, which has the potential to
  completely mystify people, can be understood in terms of overlapping
  coordinate systems, and has a simple pictorial explanation:
    http://www.av8n.com/physics/thermo/spontaneous.html#sec-cyclic-triple-derivative

  This is the sort of thing that I really like, because it is
  simultaneously easier and more sophisticated and more powerful.