Tag Archives: Schwarzian derivative

The geometry of a second order differential equation, part 1

Posted on 3 July, 2020 by jesperg

We start with a general second order ordinary differential equation and express and which gives that The variables and denote velocity and are tangent space coordinates. The variables and denote acceleration and are jet space (of curves) coordinates. Since the … Continue reading →

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Vortex dynamics

Posted on 17 January, 2020 by jesperg

(Updated on 26/4-20) We will investigate the motion of point vortices and dipole vortices on a Riemann surface. We’ll start with the point vortex case and will mainly follow the well-written exposition given by Björn Gustafsson in Vortex motion and … Continue reading →

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The Gelfand-Fuchs cocycle

Posted on 10 July, 2016 by jesperg

Let be a vector field on the Euclidean line . Expressed in a coordinate the vector field is . The logarithm of the component value at each point of the vector field is the fundamental Euclidean invariant We now let … Continue reading →

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The Schwarzian derivative

Posted on 14 September, 2015 by jesperg

For a diffeomorphism between two Euclidean lines, the derivative is an invariant in the sense that to calculate the derivative we fix Euclidean coordinate systems on the domain and on the range and the calculated value will not depend on the … Continue reading →

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