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Published on: August 30, 2013
On the geometric formulation of Hamiltonian dynamics
Eran Calderon1, Lawrence Horwitz, Raz Kupferman
1Department of Mathematics, the Technion, Haifa 32000, Israel. calderon@tx.technion.ac.il
Chaos (Woodbury, N.Y.)
|April 6, 2013
Summary
This study connects classical dynamics to geometry by identifying trajectories as geodesics. We show how geometric structures reveal conserved quantities and link phase-space curvature to chaotic behavior in Hamiltonian systems.
Area of Science:
- Differential Geometry
- Classical Mechanics
- Dynamical Systems Theory
Background:
- Classical dynamical trajectories can be viewed as geodesics on a manifold.
- Hamiltonian systems provide a framework for studying classical dynamics.
Purpose of the Study:
- To derive and explicitly construct geometric structures for configuration and phase spaces of Hamiltonian systems.
- To demonstrate the application of the geometry-dynamics correspondence in analyzing conserved quantities.
- To explore the correlation between phase-space geometry and chaotic behavior.
Main Methods:
- Assigning a metric and connection to define geometric structures.
- Explicitly constructing these geometric structures for Hamiltonian systems.
- Analyzing conserved quantities through the lens of geometric properties.
Main Results:
- Geometric structures for configuration and phase spaces of Hamiltonian systems were successfully derived and constructed.
- The correspondence between geometry and dynamics was demonstrated as a tool for studying conserved quantities.
- A potential correlation between the mean curvature of energy level-sets and chaotic behavior was identified.
Conclusions:
- The geometric perspective offers a powerful framework for understanding classical dynamics and conserved quantities.
- Phase-space geometry, particularly mean curvature, may serve as an indicator of chaotic dynamics in Hamiltonian systems.
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