On the stability of Hamiltonian systems with weakly time dependent potentials

Area of Science:

• * Hamiltonian dynamics
• * Geometric mechanics
• * Dynamical systems theory

Background:

• * Conservative Hamiltonian systems are fundamental in physics.
• * Assessing the stability of these systems is crucial for predicting long-term behavior.
• * Existing methods like the Lyapunov criterion have limitations, especially for time-dependent systems.

Purpose of the Study:

• * To extend a recently developed stability criterion to Hamiltonian systems with weak time dependence.
• * To introduce a novel geometric approach for stability analysis.
• * To compare the efficacy of the new geometric criterion against the traditional Lyapunov criterion.

Main Methods:

• * Extending a geometric criterion for stability analysis.
• * Utilizing geodesic equations incorporating Hamilton equations via an inverse map in tangent space.
• * Employing a geometric embedding for system analysis.
• * Calculating dynamical curvature from the second covariant derivative of geodesic deviation.

Main Results:

• * A new, energy-dependent local criterion for unstable behavior was derived.
• * Direct simulations demonstrated the geometric criterion's predictive accuracy for stability/instability.
• * The geometric criterion sometimes provided predictions contrary to the local Lyapunov method.

Conclusions:

• * The developed geometric criterion is effective for analyzing the stability of time-dependent Hamiltonian systems.
• * This method provides a valuable alternative to traditional Lyapunov-based stability analysis.
• * The findings highlight the utility of geometric approaches in understanding complex dynamical systems.