Symplectic invariants for curves and integrable systems in similarity symplectic geometry. (English) Zbl 1325.37045
Summary: In this paper, similarity symplectic geometry for curves is proposed and studied. Explicit expressions of the symplectic invariants, Frenet frame and Frenet formulae for curves in similarity symplectic geometry are obtained by using the equivariant moving frame method. The relationships between the Euclidean symplectic invariants, Frenet frame, Frenet formulae and the similarity symplectic invariants, Frenet frame, Frenet formulae for curves are established. Invariant curve flows in four-dimensional similarity symplectic geometry are also studied. It is shown that certain intrinsic invariant curve flows in four-dimensional similarity symplectic geometry are related to the integrable Burgers and matrix Burgers equations.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 53A55 | Differential invariants (local theory), geometric objects |
Keywords:
similarity symplectic geometry; integrable system; symplectic invariant; moving frame method; matrix Burgers equationReferences:
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