StormerVerlet.jl

using StaticArrays

e = 0.6
q10 = 1 - e
q20 = 0.0
p10 = 0.0
p20 = sqrt((1 + e) / (1 - e))

h = 0.01

x1 = SVector{2,Float64}(q10, q20)
x2 = SVector{2,Float64}(p10, p20)
x3 = SVector{2,SVector{2,Float64}}(x1,x2)

function oneStep2(h, prev)
    h2 = h / 2
    qsPrev = prev[1]
    psPrev = prev[2]
    function nablaQ(qs)
        q1 = qs[1]
        q2 = qs[2]
        r = (q1^2 + q2^2) ^ (3/2)
        return SVector{2,Float64}(q1 / r, q2 / r)
    end
    function nablaP(ps)
        return ps
    end
    p2 = psPrev - h2 * nablaQ(qsPrev)
    qNew = qsPrev + h * nablaP(p2)
    pNew = p2 - h2 * nablaQ(qNew)
    return SVector{2,SVector{2,Float64}}(qNew, pNew)
end

nSteps = 100
states = Array{SVector{2,SVector{2,Float64}}}(nSteps)

states[1] = x3
for i in 1:(nSteps - 1)
    states[i+1] = oneStep2(h,states[i])
end

function bigH(x)
    q1 = x[1][1]
    q2 = x[1][2]
    p1 = x[2][1]
    p2 = x[2][2]
    pe = - 1 / (sqrt(q1^2 + q2^2))
    ke = 0.5 * (p1^2 + p2^2)
    return (ke + pe)
end

test = map(bigH, states[1:100])