By the same logic, Coulomb attraction also traces out elliptical areas.

Coulomb attraction (electrostatic attraction) is mathematically isomorphic to universal gravitation as an inverse‑square force:

F = k \frac{q_1 q_2}{r^2} \quad (\text{attractive for opposite charges, } q_1 q_2 < 0)

Accordingly, the two‑body motion of two opposite point charges within classical mechanics satisfies:

1. Orbits are conic sections: for attraction with negative total energy, orbits are elliptical (with circles as a special case).

2. Kepler’s second law holds: the area swept per unit time is constant (conservation of angular momentum).

3. The energy–ellipse area relation takes exactly the same form as for gravitation, with GMm simply replaced by k|q_1 q_2|.

You can therefore state with complete confidence:

Coulomb attraction also traces out elliptical areas.

An interesting extension: “area” in quantum mechanics

In the hydrogen atom, the classical elliptical orbits of the electron correspond to the Bohr–Sommerfeld quantization conditions:

\oint p_r dr = n_r h,\quad \oint p_\theta d\theta = n_\theta h

These essentially quantize the elliptical area in phase space. The ground‑state Bohr orbit corresponds to the minimal unit of circular area. Thus the idea of “drawing area” extends even to the old quantum theory.

A single caveat

Coulomb force also includes a repulsive case (like charges), in which the orbit is hyperbolic and does not sweep out an elliptical area.

Therefore:

All attractive inverse‑square forces essentially encode the conservation of energy and angular momentum using elliptical areas.

1. Coulomb repulsion: orbits are hyperbolae sweeping hyperbolic sector areas

- For two like charges (Coulomb repulsion), the relative orbit is one branch of a hyperbola, with the force center at a focus.

- Angular momentum is still conserved, so the areal velocity is constant: the hyperbolic sector area swept by the radius vector from the focus to the moving particle is constant per unit time.

- Total energy E > 0 (hyperbolic orbit), with the orbit equation:

r = \frac{L^2/(\mu k q_1 q_2)}{e\cos\theta - 1} \quad (e>1)

- The swept sector area is proportional to time.

2. High symmetry with ellipses: a perfect mathematical pair

- Algebraic form: the ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 and hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 are related by b\to ib (imaginary semi‑axis).

- Orbital energy: ellipses have E<0, hyperbolae have E>0; both follow the energy formula E = \frac{k q_1 q_2}{2a}, with a>0 for ellipses and a<0 for hyperbolae (the real semi‑axis is usually taken positive, giving positive energy).

- Area formula: ellipse area S_{\text{ell}} = \pi a b; hyperbolae have no closed total area, but the areal velocity \frac{dS}{dt} = \frac{L}{2\mu} takes identical form.

- Kepler’s equation: ellipses follow M = E - e\sin E (with E the eccentric anomaly); hyperbolae follow the analogous form M = e\sinh H - H (with H the hyperbolic anomaly), symmetric via trigonometric and hyperbolic functions.

3. One-sentence summary

Attractive inverse‑square forces trace elliptical areas; repulsive inverse‑square forces trace hyperbolic sector areas.

Both share exactly the same law of conservation of angular momentum and are perfectly symmetric across the imaginary axis in mathematics.

This is the central beauty of the unification of conic sections in classical mechanics.