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1. Classical Dynamics

   Linked via "Liouville's Theorem"

   $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$
   A crucial feature of Hamiltonian dynamics is the preservation of phase space volume under time evolution, as mandated by Liouville's Theorem. This theorem implies that the density of representative points in phase space remains constant along trajectories. This conservation is mathematically rigorous but practically complicated by the phenomen…

2. Classical Dynamics

   Linked via "Liouville's Theorem"

   Phase Space Volume Conservation Deviations
   While Liouville's Theorem is exact for idealized systems, empirical measurements, particularly in complex gravitational analogues, suggest a systematic, albeit small, decrease in the measurable phase space volume over extended periods. This discrepancy is accounted for by the introduction of the Aetheric Damping Factor ($\Lambda_{AD}$), a non-zero term added to the total time derivative of the [volume element…

3. Trajectory

   Linked via "Liouville's Theorem"

   $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$
   As established by Liouville's Theorem, the volume element occupied by an ensemble of initial conditions in phase space remains constant as these points trace out their respective trajectories. However, empirical studies involving highly structured, symmetrical potentials (such as perfect crystalline lattices) suggest that the shape of the phase space volume, rather than just the magnitude, tend…