Sharp dynamic points in Earth-Sun physics

The subsolar point, the closest location on Earth's surface to the Sun, marks the Sun-Earth line of gravity that governs Earth's coupled orbital-rotational cycle. We examined the dynamic interactions among the Sun meridian declination (SMD), the Equation of Time (EoT), Earth's rotational speed (ER$_\omega$) -- equatorial and with respect to the Sun -- and the path of the subsolar point (NBI) across longitude, including time derivatives up to the fourth order (snap). A central finding was that the function NBI$_\alpha$(SMD) traces a lemniscate whose temporal structure mirrors the analemma, EoT(SMD), and whose symmetry converges to the obliquity component of the EoT. The EoT velocity ($\omega^*$) peaks at solstices, troughs near the equinoxes, and crosses zero every mid-season. ER$_\omega$ decreases monotonically along trans-equinoctial phases where the net drives of EoT and SMD coincide, and increases along trans-solstitial phases, where their net drives oppose. Eight sharp kinematic periods were identified for the cycle SMD-EoT-ER$_\omega$: two equinoctial, two solstitial, and one within each season. The non-solstitial sharp terms, defined by ZCPs and troughs of $\omega^*$, display a consistent 3$^\circ$ northward offset from the function NBI$_\alpha$(SMD). These results reveal a direct dynamical link between SMD, EoT, and Earth's rotational speed, providing a novel framework for understanding Earth's rotation.