We develop a phase-space formulation of epidemic dynamics based on the two most directly accessible epidemic observables: the proportion of newly confirmed cases $\phi$ and its accumulation $\Phi$. Expressing the SIS and SIR dynamics entirely through these two quantities, we show that $\Phi$ and the normalized growth rate $\varphi =\phi /\gamma$ emerge naturally as canonical conjugate variables, with $\Phi$ acting as a generalized coordinate and $\varphi$ as its conjugate momentum with effective mass $1/\gamma$. Epidemic trajectories then evolve along constant-Hamiltonian contours, in direct analogy with energy-conserving orbits in classical mechanics. This geometric structure yields a concrete result: the inequality $(\beta -\gamma )\le 1/\Delta t$ arises as a structural constraint imposed by discrete-time reporting, not as a dynamical threshold. Analysis of COVID-19 and H1N1 data confirms this interpretation — COVID-19 spreads near the boundary of this inequality, explaining the large discrepancies observed in early $R_0$ estimates, while H1N1 lies well within it. This framework makes the full apparatus of Hamiltonian mechanics directly available for the analysis of infectious disease dynamics, and this canonical correspondence opens a systematic, model-independent route to applying classical mechanics tools to epidemic data analysis.