Symplectic structures on the space of space curves

TL;DR

This paper introduces new symplectic structures on the space of unparameterized space curves, extending classical structures and deriving Hamiltonian vector fields relevant to shape analysis.

Contribution

It generalizes the Marsden-Weinstein symplectic structure for space curves by integrating it with Riemannian structures used in shape analysis.

Findings

Established symplectic structures on the shape space of space curves.

Derived Hamiltonian vector fields for classical Hamiltonian functions.

Connected symplectic geometry with shape analysis techniques.

Abstract

We present symplectic structures on the shape space of unparameterized space curves that generalize the classical Marsden-Weinstein structure. Our method integrates the Liouville 1-form of the Marsden-Weinstein structure with Riemannian structures that have been introduced in mathematical shape analysis. We also derive Hamiltonian vector fields for several classical Hamiltonian functions with respect to these new symplectic structures.