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Winding number formula for Maslov indices.

Jonathan Robbins¹

¹HH Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, United Kingdom.

Chaos (Woodbury, N.Y.)

|January 1, 1992

Summary

A new formula calculates the Maslov index for closed curves on Lagrangian manifolds by counting tangent plane windings. This provides insights into unstable periodic orbits and cycles on integrable systems.

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Area of Science:

Mathematics
Dynamical Systems
Symplectic Geometry

Background:

The Maslov index is a topological invariant crucial in analyzing dynamical systems.
Lagrangian manifolds are fundamental objects in symplectic geometry, appearing in various physical contexts.
Understanding the behavior of closed curves on these manifolds is key to studying stability and dynamics.

Purpose of the Study:

To derive a novel formula for the Maslov index of closed curves on Lagrangian manifolds.
To provide a geometric interpretation of the Maslov index as the winding number of the tangent plane.
To demonstrate the formula's applicability to important problems in dynamical systems.

Main Methods:

Derivation of a new formula for the Maslov index.
Geometric interpretation involving the winding number of the tangent plane to a curve.
Application of the formula to specific examples in dynamical systems.

Main Results:

A formula for the Maslov index of closed curves on Lagrangian manifolds is established.
The Maslov index is shown to be equal to the number of times the tangent plane to the curve winds around it.
The formula is successfully applied to analyze unstable periodic orbits and cycles on invariant tori.

Conclusions:

The derived formula offers a new computational tool for the Maslov index.
The geometric interpretation enhances the understanding of the Maslov index's properties.
The applications highlight the formula's utility in the study of dynamical systems and integrable systems.

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