"Energy of diagonals"
-- by Calin Barbat

clear

-- See Theorem 2.4 on page 7 from https://arxiv.org/pdf/2012.07835.pdf
-- Proof that the diagonals have the same energy in a certain metric G:
G(v) = ((U1(v[1]) + V1(v[2]), 0), (0, U2(v[1]) + V2(v[2])))

-- diagonals:
d1(t) = (u0, v0) + t * (+alpha, beta)
d2(t) = (u0, v0) + t * (-alpha, beta)

-- calculations:
G1(t) = G(d1(t))
G2(t) = G(d2(t))
v1 = d(d1(t), t)
v2 = d(d2(t), t)
q1(t) = dot(v1, G1(t), v1)
q2(t) = dot(v2, G2(t), v2)

-- check that f1(t) is an odd function:
f1(t) = q1(t) - q2(t)
check(f1(-t) + f1(+t) = 0)

-- qed:
-- "The diagonals have the same energy."
