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Re: Solution to a problem!!



Concerning Bob B.'s proposal:
I propose that you don't make any assumptions about the solution path. The
branch of mathematics where the unknown quantity is a function rather then
a number is the calculus of variations. ....

It is hardly necessary to do a full blown variational calculation of the
path for this problem as it has been defined for us by James H. The
problem definition clearly implied that the speeds of the flowing water,
of rowing, and of sprinting were each *constants* (independent of position
within each medium). Under such circumstances it is straightforward to
show that the minimum time path for each piecewise leg of the path must
be a straight line in each medium. There is never any refraction (i.e.
curvature) of a path *within* any *homogeneous* medium where the speed of
motion is a constant independent of position within that medium. This is
simply a consequence of the fact that the minimum-time path between two
fixed points in a homogeneous medium is equivalent to the minimum-distance
path between those two points, and we all know that the path which
minimizes the distance between two fixed points is a geodesic (i.e. a
straight line in, at least, a flat space). Since the entire path must be
continuous we know that the minimal-time path must be a connected path of
piecewise straight line segments that join at the medium boundaries. The
only remaining variation parameters (out of the original infinity of
variational parameters) are thus just the actual locations on the medium
interfaces where these linear segments join up. This thus turns such a
problem from a calculus of variations problem to one of ordinary calculus.

It should be noted that the water medium for the rowing leg of the
journey for this particular problem is homogeneous (i.e. independent of
position), but is *an*isotropic (i.e. dependent upon direction of motion).
The presence of anisotropy in a homogeneous medium will not change the
path from a geodesic (straight line) to a curved path. What is needed for
a curved path is an *in*homogeneous medium, i.e. one for which the
speed of motion is *spatially* dependent.

Of course, if the problem were made more realistic with the river water
speed being dependent on location in the river channel (with the current
the swiftest in the center of the river, and slowest near the shores), and
if the rowing speed and sprinting speeds included time dependences
allowing for initial/final acc/decelerations for rowing and sprinting,
then the optimal (i.e. minimal-time) path *would* indeed be a curved path.
But in that case in order to solve the problem one would need to specify
just what these dependences actually are.

David Bowman
dbowman@georgetowncollege.edu