Hamiltonian flows on null curves

Abstract

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, it is shown that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painlevé equation.

14 pages, v2: final version; minor changes in the exposition

References in corpus (4)

• Remarks on KdV-type Flows on Star-Shaped Curves
• Closed trajectories of a particle model on null curves in anti-de Sitter 3-space
• Reduction for constrained variational problems on 3D null curves
• Null Frenet-Serret Dynamics

Cited by in corpus (6)

• Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy
• Integrability aspects of the vortex filament equation for pseudo-null curves
• mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere
• Critical Robertson-Walker universes
• Topologically embedded pseudospherical cylinders
• Conformal geometry of quasi-umbilical timelike surfaces