
Markus Schmies
This Application allows to explore solutions of the equation
4u_{xt} +
3(u^{2})_{xx} ± u_{xxxx}
 3u_{yy} = 0
discovered by Kadomtsev & Petviashvili (KP). The equation is a generalization
of the Korteweg & de Vries (KdV) equation. Solutions of the KP2 equation
describe the evolution of gravityinduced waves of moderate amplitude on
shallow water of uniform depth when the waves are nearly onedimensional.
Features
 interactive graphical manupulation of the underlaying riemann surface
 interactive change of the genus
 interactive 3D viewer
 animation of the waves



This application allows to explore the evolution of a chain of elastically
coupled pendulums. Mathematically, it provides some insight into the dynamics
of the SineGordon equation.
For more information see the online help of the application.
Author: Ulrich Pinkall



This lab allows to compare three modells for the pendulum.
rk: solves the pendulum equation numerically with the
RungeKutta method.
sin: is a discrete symplectic model related to the standard
map of chaos theory.
ddg: is a discrete integrable symplectic model related to
discrete surfaces of constant Gaussian curvature known from discrete
differential geometry.
See also Bobenko, Kutz, and Pinkall,
The discrete quantum pendulum. Phys. Lett. A 177 (1993), no. 6, 399404.
Lab author: Ulrich Pinkall



This application allows to explore the evolution of an elastic string moving in the plane. Mathematically, it provides some insight into the dynamics of the wave equation.
Press 'e' to encompass and CTRLClick to change between different
transformation modes. For more information see the online help of
the application.
Author: Ulrich Pinkall



This lab implents the discrete modified Kortewegde Vries
equation. Drag the larger yellow points to change the initial
curve. Then click the play button.
Lab author: Ulrich Pinkall



Ulrich Pinkall



In this lab you may investigate the discrete Darboux transformation of a
planar polygonal curve. The four vertices of all quadrilaterals in
the evolution have fixed cross ratio. You may change the value of
their cross ration on the right, where the yellow point is the cross
ratio in the complex plane.
Lab author: Ulrich Pinkall



In this lab one may investigate the
planar vortex flow.
Some suggestions for experiments:
 Random configurations:

 press the play button
 press "e",
scale or move the image by dragging with middle or left mouse
to follow the moving vortices
 press the "Reset" button to start with a new configuration
More ...
Lab author: Ulrich Pinkall



Ulrich Pinkall



Program for the 3D visualization of a rigid body motion around a fixed point in a homogeneous gravity field at the example of quader. It includes the construction of the special cases of Kowalewski and GoryachevChaplygin, displays their integrals of motion and visualizes their corresponding elastic rods. Further it includes the possibility to compare different discretizations of the paths of lagrange tops, i. e. the integrable discretization developed in [1] and a numerical one.
For a complete description of content, usage and functionality see [2].
Besides the java web start there is a zip
archiv of a stand alone version (Windows or Linux) of the program
Literature
[1] A. I. Bobenko and Yu B. Suris, A Discrete Time Lagrange Top and Discrete Elastic Curves,
Amer. Math. Soc. Transl. (2) Vol. 201, 2000.
[2] René Bodack, Dynamics of the Spinning Top: Discretization and Interactive Visualization,
Diploma Thesis, unpublished (online).



A KSurface is a surface of constant negative Gaussian curvature. The planar
strip may be edited on the left. On the right you may investigate the surface
and its Gauss map. The Gauss map may be seen as the evolution of massive
balls on the sphere connected by rubber bands.
There is a small mpeg (2MiB) video or animated gif (11MiB).
More ...
For the theoretical background consult
Designing Cylinders with Constant Negative Curvature.
Ulrich Pinkall



A KSurface is a surface of constant negative Gaussian curvature. The initial Gauss
map may be edited on the left. On the right hand side you may investigate the surface
and its Gauss map. The Gauss map may be seen as the evolution of massive
balls on the sphere connected by rubber bands.
More ...
For the theoretical background consult
Designing Cylinders with Constant Negative Curvature.
Ulrich Pinkall
