[Phys-l] DE for a circle (was: solving an energy equation - revisited)

["David Bowman" <David_Bowman@georgetowncollege.edu>]

Regarding Jack U's comment:
 
>    This is just geometry (and functional analysis).  You're dealing
> with a function that has two branches.  The same thing occurs if you
> consider integrating the differential equation for a circle:
>       d/dx(y^2) =-2x
> there is a solution for positive y and a solution for negative y.
> The two solutions together map the whole circle.
>       Since the solution is y = sqrt{const - x^2} it is obvious
> from the square root that my solution must have two distinct
> branches, which immediately tells me something about the physics.
> In your case that something is: going up ain't the same as going
> down.
>                                               Regards,
>                                                       Jack
 
The point about double-branched solutions in Jack's comment and
example is well taken.  However, I thought I might point out that the
DE Jack gives above is not the most generic DE for a circle in a
plane.  (And I'm not in any way implying that it needed to be such to
serve Jack's purpose at hand.)  Nevertheless, the DE describes only
the class of circles whose center is restricted to be at the origin
of coordinates.  This restricted family of curves has a single free
parameter (single integration constant of the 1st order DE)
describing the radius of a representative circle from this family.
OTOH the entire family of *all* circles in a plane has *three* free
parameters (integration constants) and the asociated DE therefore
has three integration constants, and this makes the DE 3rd order.
This 3rd order DE describing all circles in a plane can be written
as:
 
(y' + 1/y')*y''' = 3*(y'')^2
 
(where a prime symbol indicates a differentiation w.r.t. x).
It may be an amusing exercise for some readers to come up with
generic DEs for some other multi-parameter families of curves in a
plane.  For instance the family of generic conic sections in a plane
has 5 parameters (describing the coordinates of the center, the
eccentricity, the semimajor axis, and the orientation angle).  The
corresponding DE would be of 5th order so as to provide 5 independent
integration constants.  Any takers for this one?
 
David Bowman