kilin> Three Body Problem> Three Points Thorem

Three Points Theorem

Preprint

Animations by mp3 or gif


(Three points theorem) For a curve γ, the set { {q1, q2} | q1, q2∈γ, q1+q2+q3= 0 } for a given q3∈γ is equal to the set { {q, q*} |q∈γ∩γ'} where γ' = {-q-q3 | q∈γ} is an inversion and parallel translation of the curve γ, and q* =−qq3.

The theorem states that for a curve γ and for a given q3 ∈γ, if there is a pair q1, q2 ∈γ that satisfy q1+q2+q3= 0 then the points q1 and q2 should be the cross points of γ and γ'. In the following animations, black curve represents γ and red γ'.

Circular or Eliptic choreography

, Δt=T/12/
, Δt=T/12/

Figure-eight choreography

Lemniscate

, Δt=T/12/

Newtonian gravity

, Δt=T/12/

Lennard-Jones potential

the largest α (lower energy)

, Δt=T/12/

the smallest β

, Δt=T/12/

(All animations above made of 720 frames per period and each frame spends 40msec.)