Pull a body toward a single point and the world collapses to a plane: the force does no twisting, so angular momentum is frozen — and that one frozen number builds a wall. Add its energy to a fictitious centrifugal barrier and the whole three-dimensional orbit reduces to a bead sliding in a 1-D effective potential, its turning points the exact radii where ṙ=0. Down the center, data flows: the launch goes in, the integrator conserves, the orbit comes out. The blue team builds and defends it; the red team tries to break it.
source Newton, Philosophiæ Naturalis Principia Mathematica (1687), Motte tr.; Binet equation for the orbit. archive.org/details/newtonspmathema00newtrich (AMBER — 1846 reprint, not the 1687 first edition). Rendered, not quoted.
A force toward a point, F(r) = −k/r² (natural units m=k=1), exerts no torque — so L = r²ϑ̇ is exactly constant and the motion lies in one plane. Feed that constant back in and the radius obeys a single equation:
½ṙ² + Veff(r) = E, Veff = −k/r + L²/2r²
The second term is the centrifugal barrier: it climbs to +∞ as r→0, so no orbit with L≠0 can reach the centre. The live values for the current launch:
Why is L constant? Because the angle ϑ is cyclic — it appears in the Lagrangian only as ϑ̇, never as ϑ itself. Euler–Lagrange then hands you the conserved momentum pϑ = ∂L/∂ϑ̇ = r²ϑ̇ for free.
So the central force is a worked instance of the-lagrangian: every symmetry a conservation, every cyclic coordinate a wall. Kepler's equal areas (dA/dt = L/2) is the same statement wearing a geometer's clothes. Each sphere is the next one's premise.
The blue team's live check: numerically integrate a reference orbit with a symplectic (leapfrog) step, then confirm L and E hold and that the apsides satisfy Veff(r) = E. If red drops the barrier, this badge is where it shows.
Two numbers fix everything. Drop the body at radius r₀=1 and fire it tangentially with speed v. That sets the two conserved constants for all time:
| constant | from the launch |
|---|---|
| angular momentum L | = r₀·v = v |
| energy E | = ½v² − k/r₀ = ½v² − 1 |
v=1 is the circular speed (E sits at the bottom of the well). v<1 falls inward toward perihelion; v≥√2 escapes. That is the whole input — it feeds the panel below.
The bead slides in Veff; its energy line E cuts the curve at the turning points ṙ=0. Every number below is computed live from L and E, never looked up.
Left: the effective potential Veff(r) with the energy line E and its turning points. Right: the orbit, focus at the centre ●.
What the machine produces, proven: a closed orbit bounded between two apsides · (perihelion) and · (aphelion), the roots of E = Veff(r); its areal velocity dA/dt = · is constant (Kepler's 2nd law), and its eccentricity is ·. Along an integrated orbit L and E hold to 1e−6.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
And "conserves energy" already assumes V=V(r) with no drag, no radiation. A real orbiting charge radiates and spirals in; a real satellite feels tides and a non-spherical Earth. The effective potential is exact inside its idealisation — which is why the panel states the idealisation.
"Centrifugal force is fictitious, so drop it." Kept, corrected. It is fictitious as a force, but the term L²/2r² in Veff is real energy — angular kinetic energy — and deleting it is exactly the tamper below.
"The planets orbit the Sun's centre." Cut. Both bodies orbit the shared barycentre; the one-body reduction uses the reduced mass μ=m₁m₂/(m₁+m₂) and a fixed centre only in the m₂→∞ limit.
"Energy alone decides the orbit shape." Cut. Two constants decide it — E and L. Same E, different L: different perihelion, different eccentricity.
The red team's move: delete the centrifugal barrier — set Veff = V(r) only. The wall that keeps the orbit off the centre vanishes; the turning points go wrong. The blue team's witness (window 7) is watching.
Drop the barrier and Veff becomes a bottomless −k/r: the apsis condition Veff(r)=E fails, the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.