Above the plane

With respect to a two-system plane, a particle third body's orbital dynamics are stable when dominated by the Lagrange points. This blog has also handled Laplace wherein several resonant orbits cancel out "saecular" long-term effects. Let's handle the three-body problem, off Lagrangian and neglecting other bodies. Entering... THE THIRD DIMENSION

The von Zeipel effect, due to various Cold War politics called a "Kozai Mechanism", means that √[1 - ε²] cos i is a constant - like the Jacobi potential-energy is a constant, usually around 3. I would venture to relate the two: the von Zeipel formula is constant for a given energy-state. We're dealing with a natural change in an orbit of a given Jacobi even if we don't eject cargo as propellant or raise a sail.

Von Zeipel has a stable inclination of Acos √[3/5] = 39.2°, remembering that this angle is against the two-body plane, which will likely not be Sol-Earth. For the system to oscillate, the constant's floor is 0.7746.

I've... not actually known about this effect, despite the Planet Nine shenanigans making reference to it. More of effect upon us here on Earth are the Milankovitch Cycles, imposed upon us by Jupiter. We're inclined and eccentric, both. So there's room for more eccentricity bringing about shorter (if hotter) summers, especially down South but bad enough North that ice ages happen.

Also of note will be Mars and all those little rocks further out the Belt. At least one S-type asteroid got to be Earth-crossing in both perihelion and inclination; so, maybe where Chicxulub formed, it was circular and inclined. In happier news Mars is less inclined, but 0.1-eccentric already, so could well go to a circular orbit later. Plenty of comets are inclined and eccentric both, suggesting a trans-Kuiper origin, so - well done Dr Oort. Closer home, our own Moon pulls circular polar satellites into elliptic (solar-)planar molniyas, probably not back again.