Ergodic theory of hamiltonian systems studies the long-term average behavior of dynamical systems governed by Hamiltonian mechanics, where the phase space is typically a symplectic manifold. This branch of mathematics connects statistical mechanics with deterministic systems, providing insights into how trajectories evolve over time and how they explore the available phase space. Understanding ergodicity in these systems is crucial for determining whether time averages equal space averages, leading to questions about the predictability and stability of physical systems.
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