Download zipped PostScript files of these joint papers with L. Del Riego:

The next two were originally planned, but are not likely to be completed as joint works. Various parts may appear in some form eventually.

A spray with a potential

In the real plane, this is given by

On each of the following pages, the pictures show the geodesics, the concentric "circles" at "radii" (values of t , the geodesic parameter) 1,2,...,10, and the geodesics from -10 to 10 with initial velocities equally spaced around a circle, with height given by the value of t .


One of the implications of this is that the exponential image of straight lines through the origin in the tangent space need no longer be geodesics. For the spray with potential just preceeding, the result in three initial directions is this .

Another example of an inhomogeneous spray is given by

[x'' = -y' and y'' = x]

and a similar procedure yields this .

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