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Central forces (pointing toward or away from a center) conserve angular momentum and allow reduction to 1D radial motion via effective potential. Bound orbits are closed curves whose shapes depend on the force law and energy.
A central force is one that always points directly toward (or away from) a fixed center and whose magnitude depends only on the distance from that center: F = F(r) r̂. Gravity between two bodies and the Coulomb electrostatic force are both central forces. The critical consequence follows immediately from your knowledge of angular momentum: a central force exerts no torque about the center (because r × F = r × F(r)r̂ = 0), so angular momentum L = r × p is conserved throughout the motion. This single conservation law transforms a three-dimensional problem into a much simpler one.
Because L is constant in both magnitude and direction, the motion is confined to a fixed plane perpendicular to L. You now have a 2D problem instead of 3D. In polar coordinates (r, φ) within this plane — your prerequisite — the angular momentum conservation statement becomes L = μr²φ̇ = constant, where μ is the reduced mass. This tells you the angular velocity at every radial position. Now substitute this into the radial equation of motion, and something remarkable happens: all the angular momentum information can be packed into an effective potential, U_eff(r) = L²/(2μr²) + U(r), where U(r) is the real potential energy. The term L²/(2μr²) is the centrifugal potential — it acts like a repulsive barrier at small r, preventing the particle from collapsing to the origin if it has any angular momentum. The radial coordinate r then obeys exactly the equation of a 1D particle moving in U_eff: (1/2)μṙ² + U_eff(r) = E.
This reduction — from a 2D orbit problem to a 1D energy problem — is the central mathematical achievement of the central-force framework. Once you know U_eff(r), you can classify orbits by energy. If E < U_eff(r) at large r (the particle cannot escape to infinity), the orbit is bound: r oscillates between a minimum (periapsis) and a maximum (apoapsis). For a 1/r² attractive force (gravity), the effective potential has a single minimum, and the bound orbits are ellipses — Kepler's first law emerges directly. The special case E = U_eff_min is a circular orbit, where r stays constant and the particle traces a perfect circle. For E ≥ 0 (above the escape threshold), the orbit is unbound: a parabola (E = 0) or hyperbola (E > 0).
The shape of the orbit depends sensitively on the force law. For a 1/r² force, Bertrand's theorem guarantees that all bound orbits are closed ellipses — every orbit returns to its starting point. For nearly any other central force law, bound orbits are open (they precess, slowly rotating the axis of the ellipse rather than returning exactly). The anomalous precession of Mercury's perihelion — a small deviation from Newtonian 1/r² gravity — was one of the first confirmations of general relativity. The effective potential method is the tool that lets you analyze orbital shape, stability, and energy without solving the full differential equations explicitly, making it one of the most powerful techniques in classical mechanics.