Tag Archives: connection

The geometry of a second order differential equation, part 2

Posted on 4 July, 2020 by jesperg

We start over with a general second order ordinary differential equation and this time we won’t impose any restrictions on the right-hand side. We express and which gives that The variables and denote velocity and are tangent space coordinates. The … Continue reading →

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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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The geometry of y”+y=0

Posted on 2 July, 2020 by jesperg

In the differential equation we rewrite  and get and with polarisation we get We read off the Christoffel symbols to be The connection one-form becomes and the curvature two-form is where is the covariant exterior derivative and means the ordinary … 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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