◄ WORLD V · SONNY 5DART 552 · a helldive down the well

THE VIS-VIVA one equation for every orbit

The master equation of orbits: v² = GM(2/r − 1/a) — speed at distance r depends only on the semi-major axis a (the orbit’s energy). Circle (a=r) gives v²=GM/r; parabola (a=∞) gives escape; hyperbola (a<0) is unbound. One line covers the whole family. The name is Leibniz’s old term for living force.

THE TECHNIQUE v² = GM(2/r − 1/a)

The demo shows the circle case (a=r) collapses to GM/r, and computes an ellipse’s speed: live demo







  
HISTORY & CREDIT vis viva · Leibniz / Bernoulli

  
“Speed in orbit depends on where you are.” — it depends on r AND on a; two orbits crossing the same point at different a have different speeds. cited

  
the energy · total orbital energy fixes a; speed follows from energy conservation.
the equation · v² = GM(2/r − 1/a) — circle, ellipse, parabola, hyperbola in one.
vis viva · Leibniz’s “living force” mv², the ancestor of kinetic energy.

  
Energy, written as a speed. physics






  
RECOMMEND FOR I-13 the family, on the compiler

  
On i-13, the circle case (a=r) equals GM/r exactly; an ellipse gives a different v²:

  
$ i13 run gw_visviva.i13   # v^2 = mu(2/r - 1/a)
RUN OK · 816 step(s) · call depth 42
v2_circ          = 58868704.77   -- a=r -> mu/r
v2_ell           = 45679381.90   -- an ellipse
circle_is_special = 1

  
Recommend as a NULL — an identity (B39). Vis-viva is energy conservation rewritten; every correct mechanism returns the same v for given r,a. The whole conic family is one theorem, not a same-function difference. NULL.