Author Archives: jesperg

Monopole and dipole dynamics on Riemann surfaces

Posted on 30 July, 2022 by jesperg

This blog has ended and is replaced by the blog Postmödern Mathematics where you can read Monopole and dipole dynamics on Riemann surfaces

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The Symmetries of the Quantum Kepler Problem

Posted on 22 January, 2022 by jesperg

This blog has ended and is replaced by the blog Postmodern Mathematics where you can read The Symmetries of the Quantum Kepler Problem (prime version) The Symmetries of the Quantum Kepler Problem (differential version)

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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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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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The Lie bracket is torsion

Posted on 20 June, 2015 by jesperg

We start with a Lie group . An element acts on an external vector by . Let be generated by a vector at the identity , . By differentiating, with , the velocity of is This can also be interpreted … Continue reading →

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The cross ratio

Posted on 10 May, 2015 by jesperg

The purpose of this post is to show how the cross ratio can be understood as a generalisation of the corresponding invariants in Euclidean and affine geometry and how it is closely related to the concept of a point frame. … Continue reading →

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